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Definition Curvature Curve Sketching Second Derivative Test Higher Derivatives Bernd Schröder Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. y 6 x Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. y 6 a Bernd Schröder Higher Derivatives x logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. y 6 a Bernd Schröder Higher Derivatives b x logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. y 6 v a Bernd Schröder Higher Derivatives b x logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. y 6 v v a Bernd Schröder Higher Derivatives b x logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. y 6 v v a Bernd Schröder Higher Derivatives b x logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. y 6 v v a Bernd Schröder Higher Derivatives b x logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. y 6 v v a Bernd Schröder Higher Derivatives b x logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. y 6 v v a Bernd Schröder Higher Derivatives b x logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. y 6 v v a Bernd Schröder Higher Derivatives b x logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. y 6 v v a Bernd Schröder Higher Derivatives b x logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. y 6 v v a Bernd Schröder Higher Derivatives b x logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. y 6 v v a Bernd Schröder Higher Derivatives b x logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. y 6 v v a Bernd Schröder Higher Derivatives b x logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. 2. Velocities can change, just like positions can. The rate of change of the velocity is the acceleration. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Do We Need Higher Derivatives? 1. Just because we know a function is, say, decreasing, does not mean we really know the function’s behavior. 2. Velocities can change, just like positions can. The rate of change of the velocity is the acceleration. 3. Higher order derivatives can be used to approximate functions with (Taylor) polynomials. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Definition. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Definition. Let f be a function whose derivative f 0 is differentiable, too. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Definition. Let f be a function whose derivative f 0 is differentiable, too. Then f is called twice differentiable and we 00 0 0 denote f := f . Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Definition. Let f be a function whose derivative f 0 is differentiable, too. Then f is called twice differentiable and we 00 0 0 denote f := f . In general, if f is n-times differentiable with nth derivative f (n) , then f is called (n + 1)-times differentiable Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Definition. Let f be a function whose derivative f 0 is differentiable, too. Then f is called twice differentiable and we 00 0 0 denote f := f . In general, if f is n-times differentiable with nth derivative f (n) , thenf is called (n + 1)-times differentiable if and only if f (n+1) := f (n) Bernd Schröder Higher Derivatives 0 exists. logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Definition. Let f be a function whose derivative f 0 is differentiable, too. Then f is called twice differentiable and we 00 0 0 denote f := f . In general, if f is n-times differentiable with nth derivative f (n) , thenf is called (n + 1)-times differentiable if and only if f (n+1) := f (n) 0 exists. Another notation for the nth derivative is Bernd Schröder Higher Derivatives dn f (x) := f (n) (x). dxn logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Which Car Do You Think Will Win? Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Which Car Do You Think Will Win? s 6 t Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Which Car Do You Think Will Win? s 6 finish line t Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Which Car Do You Think Will Win? s 6 finish line t Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Which Car Do You Think Will Win? s 6 finish line u t Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Which Car Do You Think Will Win? s 6 finish line u car 1 t Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Which Car Do You Think Will Win? s 6 finish line u car 1 car 2 t Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Definition. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Definition. A twice differentiable function f is called Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Definition. A twice differentiable function f is called concave up on the interval [a, b] Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Definition. A twice differentiable function f is called concave up on the interval [a, b] if and only if f 00 (x) > 0 for all x in [a, b]. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Definition. A twice differentiable function f is called concave up on the interval [a, b] if and only if f 00 (x) > 0 for all x in [a, b]. It is called concave down on [a, b] Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Definition. A twice differentiable function f is called concave up on the interval [a, b] if and only if f 00 (x) > 0 for all x in [a, b]. It is called concave down on [a, b] if and only if f 00 (x) < 0 for all x in [a, b]. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Definition. A twice differentiable function f is called concave up on the interval [a, b] if and only if f 00 (x) > 0 for all x in [a, b]. It is called concave down on [a, b] if and only if f 00 (x) < 0 for all x in [a, b]. A point x where the function changes from concave down to concave up or vice versa is called an inflection point. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Definition. A twice differentiable function f is called concave up on the interval [a, b] if and only if f 00 (x) > 0 for all x in [a, b]. It is called concave down on [a, b] if and only if f 00 (x) < 0 for all x in [a, b]. A point x where the function changes from concave down to concave up or vice versa is called an inflection point. An inflection point necessarily occurs where f 00 is zero or undefined. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test All Combinations of Growth Behavior and Concavity are Possible Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test All Combinations of Growth Behavior and Concavity are Possible Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test All Combinations of Growth Behavior and Concavity are Possible increasing Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test All Combinations of Growth Behavior and Concavity are Possible increasing Bernd Schröder Higher Derivatives decreasing logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test All Combinations of Growth Behavior and Concavity are Possible increasing decreasing concave up Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test All Combinations of Growth Behavior and Concavity are Possible increasing decreasing concave up concave down Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test All Combinations of Growth Behavior and Concavity are Possible increasing decreasing concave up concave down Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test All Combinations of Growth Behavior and Concavity are Possible increasing decreasing concave up concave down Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test All Combinations of Growth Behavior and Concavity are Possible increasing decreasing concave up concave down Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test All Combinations of Growth Behavior and Concavity are Possible increasing decreasing concave up concave down Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1 Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2 Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞ x→∞ Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ Bernd Schröder Higher Derivatives x→−∞ logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f0 Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f0 Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f0 Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f0 Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f0 Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f0 −1 Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f0 −1 Bernd Schröder Higher Derivatives 1 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f0 −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f 0− − − −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f 0− − − +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f 0− − − −−− +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f 0− − − −−− +++ +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f decr. f 0− − − −−− +++ +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f decr. f 0− − − incr. +++ −−− +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f decr. f 0− − − incr. +++ decr. −−− +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f decr. f 0− − − incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f decr. f 0− − − incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f 00 f decr. f 0− − − incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f 00 f decr. f 0− − − incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f 00 f decr. f 0− − − incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f 00 f decr. f 0− − − incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f 00 0 f decr. f 0− − − incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f 00 f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f 00 +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f 00 −−− +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f 00 −−− +++ +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f 00 conc. up +++ −−− +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f 00 conc. up +++ conc. down −−− +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ Second Derivative Test 6 f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ Second Derivative Test 6 - f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ Second Derivative Test 6 - x f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ 6 - x f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ 6 - x f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ 6 - x f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ 6 - x f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ 6 - x f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ 6 - x −1 f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ 6 - −1 f f 00 x 1 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ 6 - −1 f f 00 1 conc. up +++ x 2 conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ 6 - −1 f f 00 1 conc. up +++ 2 conc. down −−− x 3 conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ 6 - −1 f f 00 1 conc. up +++ 2 conc. down −−− x 3 conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ 6 - −1 f f 00 1 conc. up +++ 2 conc. down −−− x 3 conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ 6 - −1 f f 00 1 conc. up +++ 2 conc. down −−− x 3 conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ 6 - −1 f f 00 1 conc. up +++ 2 conc. down −−− x 3 conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ 6 - −1 1 2 x 3 −1 f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 1 f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 x 3 −1 f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 2 1 f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 x 3 −1 f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 1 f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 x 3 −1 f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b 1 f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 x 3 −1 f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ 6 3 2 b b 1 - −1 1 2 x 3 −1 f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 x 3 −1 f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 x 3 −1 f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 b −1 f f 00 x 3 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 b −1 f f 00 x 3 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 b −1 f f 00 x 3 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 b −1 f f 00 x 3 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 b −1 f f 00 x 3 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 b −1 f f 00 x 3 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 x 3 b −1 f f 00 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 b −1 f f 00 x 3 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 b −1 f f 00 x 3 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 2 b −1 f f 00 x 3 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 b −1 f f 00 x 3 2 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 b −1 f f 00 x 3 2 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 b −1 f f 00 x 3 2 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 b −1 f f 00 x 3 2 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 b −1 f f 00 x 3 2 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 b −1 f f 00 x 3 2 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 b −1 f f 00 x 3 2 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test y Sketch the Graph of f From the Information Below 6 3 2 b b b 1 b f 0 > 0 on (−1, 1) and on (3, ∞) f 0 < 0 on (−∞, −1) and on (1, 3) f 00 > 0 on (−∞, 0) and on (2, ∞) f 00 < 0 on (0, 2) f (−1) = 1, f (0) = 32 , f (1) = 2, f (2) = 21 , f (3) = −1 lim f (x) = ∞, lim f (x) = ∞ x→∞ x→−∞ - −1 1 b −1 f f 00 x 3 2 conc. up +++ conc. down −−− conc. up +++ f decr. f 0− − − 0 2 incr. +++ decr. −−− incr. +++ −1 Bernd Schröder Higher Derivatives 1 3 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Theorem. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Theorem. The second derivative test. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Theorem. The second derivative test. Let f be a function that is twice differentiable in an interval (a − ∆, a + ∆), so that f 00 is continuous in this interval and f 0 (a) = 0. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Theorem. The second derivative test. Let f be a function that is twice differentiable in an interval (a − ∆, a + ∆), so that f 00 is continuous in this interval and f 0 (a) = 0. 1. If f 00 (a) < 0, then f has a local maximum at a. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Theorem. The second derivative test. Let f be a function that is twice differentiable in an interval (a − ∆, a + ∆), so that f 00 is continuous in this interval and f 0 (a) = 0. 1. If f 00 (a) < 0, then f has a local maximum at a. 2. If f 00 (a) > 0, then f has a local minimum at a. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Theorem. The second derivative test. Let f be a function that is twice differentiable in an interval (a − ∆, a + ∆), so that f 00 is continuous in this interval and f 0 (a) = 0. 1. If f 00 (a) < 0, then f has a local maximum at a. 2. If f 00 (a) > 0, then f has a local minimum at a. For single variable functions, the second derivative test can often be replaced with an analysis of growth behavior. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Theorem. The second derivative test. Let f be a function that is twice differentiable in an interval (a − ∆, a + ∆), so that f 00 is continuous in this interval and f 0 (a) = 0. 1. If f 00 (a) < 0, then f has a local maximum at a. 2. If f 00 (a) > 0, then f has a local minimum at a. For single variable functions, the second derivative test can often be replaced with an analysis of growth behavior. But for multivariable functions, the higher-dimensional analogue of this theorem is just about the only way to identify extrema. Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f 00 Bernd Schröder Higher Derivatives logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f 00 Bernd Schröder Higher Derivatives f 00 < 0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f0 f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f0 f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f0 a f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f0 f 0 (a) = 0 a f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f0 ··· PP f 0 (a) = 0 PP PP a PP ··· f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f0 ··· f0 > 0 PP f 0 (a) = 0 PP PP a PP ··· f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? implies 6 f0 ··· f0 > 0 PP f 0 (a) = 0 PP PP a PP ··· f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f increasing implies 6 f0 ··· f0 > 0 PP f 0 (a) = 0 PP PP a PP ··· f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f increasing implies 6 f0 ··· f0 > 0 f0 < 0 PP f 0 (a) = 0 PP PP a PP ··· f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f increasing implies 6 f0 ··· implies 6 f0 > 0 f0 < 0 PP f 0 (a) = 0 PP PP a PP ··· f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f increasing f0 ··· f decreasing implies 6 implies 6 f0 > 0 f0 < 0 PP f 0 (a) = 0 PP PP a PP ··· f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f f increasing f0 ··· f decreasing implies 6 implies 6 f0 > 0 f0 < 0 PP f 0 (a) = 0 PP PP a PP ··· f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f f increasing f0 ··· f decreasing implies 6 implies 6 f0 > 0 f0 < 0 PP f 0 (a) = 0 PP PP a PP ··· f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f a f increasing f0 ··· f decreasing implies 6 implies 6 f0 > 0 f0 < 0 PP f 0 (a) = 0 PP PP a PP ··· f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science Definition Curvature Curve Sketching Second Derivative Test Why Does the Second Derivative Test Work? f ··· ··· a f increasing f0 ··· f decreasing implies 6 implies 6 f0 > 0 f0 < 0 PP f 0 (a) = 0 PP PP a PP ··· f 0 decreasing f 00 Bernd Schröder Higher Derivatives implies 6 f 00 <0 logo1 Louisiana Tech University, College of Engineering and Science