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Mathematical Economics Concavity and Convexity Relative Extrima Inflection Points Optimization of Functions Optimization Presented by: Nadia Batool Roll No. 8528 Rabia Naseer Roll No. 8503 Madeeha Iqbal Roll No. 8523 Mumtaz Hussain Roll No. 8506 Increasing & Decreasing Function: A function f ( x) is said to be increasing/decreasing at xa if in the immediate vicinity of the point [a, f (a)] the graph of the function rises/falls as it moves from left to right. Since the first derivative measures the rate of change and slope of a function, a positive first derivative at xa indicate that the function is increasing at s; negative first derivative indicates it is decreasing. f (a) 0 : increasing function at x a f (a) 0 : decreasing function at x a Increasing & Decreasing Function: Monotonic Function: A function that increases/decreases over its entire domain is called monotonic function. It is said to increase/decrease Monotonically . Concavity & Convexity : A function f ( x) is concave at x a if in some small region close to the point [a, f (a)] the graph of the function lies completely below the tangent line. A function is convex at x a if in an area very close to [a, f (a)] the graph of the function lies completely above the tangent line. A positive second derivative at x a denotes that the function is convex at x a ; a negative second derivative at x a denotes the function is convex at a. The sign if first derivative is irrelevant for concavity. f (a) 0 : f ( x) is convex at x a f (a) 0 : f ( x) is concave at x a Concavity & Convexity : Concavity & Convexity : Relative Extrema: f (a) O f (a) O f (a) O : Relative Minimum at x a f ( a) O : Relative Maximum at x a Inflection Points: An inflection point is a point on the graph where the function crosses its tangent line and changes from concave to convex or vice versa. Inflection points occur only where the second derivative equals to zero or is undefined. The sign of first derivative is immaterial. f (a) O or is undefined Concavity changes at x a Graph crosses its tangent line at x a Inflection Points: Optimization of a Function: Optimization is the process of finding the relative maximum or minimum of a function. It is developed through usual differential functions Step I: Take the first derivative, set it equal to zero, and solve for the critical points. This step represents the necessary condition know as the first-order condition. It identifies all the points at which the function is neither increasing nor decreasing, but at a plateau. All such points are candidates for a possible relative maximum or minimum Optimization of a Function: Step II: Take the second derivative, evaluate it at the critical points and check the signs. If at a critical point a, f (a) O, the function is concave at a, and hence at a relative maximum f (a) O, the function is convex at a, and hence at a relative minimum f (a) O, the test is inconclusive Note that if the function is strictly concave/convex there will be only one maximum/minimum called a global maximum/global minimum. Optimization of a Function: Example Example: Optimize f ( x) 2 x3 30 x2 126 x 59 (a) Find the critical points by taking the first derivative, setting it equal to zero and solving it for x f ( x) 6 x2 60 x 126 6( x2 10 x 21) 0 x 2 7 x 3x 21 0 x( x 7) 3( x 7) 0 ( x 3)( x 7) 0 x 3, x 7 Citical Points Optimization of a Function: Example (b) Test for concavity by taking the second derivative, evaluating it at the critical points, and checking the signs to destinguish between a relative maximum and minimum f ( x) 6 x2 60 x 126 f ( x) 12 x 60 At x 3 f (3) 12(3) 60 36 60 24 0 At x 7 f (7) 12(7) 60 84 60 24 0 Concave, relative maximum Convex, relative minimum The function is maximized at x 3 and minimized at x 7 Optimization: Step I: Find the critical values Step II: Test for concavity to determine relative maximum or minimum Step III: Check the inflection points Step IV: Evaluate the function at the critical values and inflection points. Example: Optimize f ( x) x3 18x2 96 x 80 Optimization: Example Optimize f ( x) x3 18x2 96x 80 Step I: Find the critical values f ( x) 3x2 36 x 96 3( x2 12x 32) 0 x 2 8x 4 x 32 0 x( x 8) 4( x 8) 0 ( x 8)( x 4) 0 x 8, x 4 Citical Points Optimization: Example Step II:Test for concavity to determine relative maximum or minimum f ( x) 6 x 36 At x 4 f (4) 6(4) 36 24 36 12 0 At x 8 f (8) 6(8) 36 48 36 12 0 Concave, relative maximum Convex, relative minimum Step III: Check the inflection points f ( x) 6 x 36 0 6( x 6) 0 x6 Inflaction Point Optimization: Example Step IV:Evaluate the function at the critical values and inflection points. At x 4 f (4) x3 18(4)2 96(4) 80 80 (4,80) Relative Maximum At x 6 f (6) x3 18(6)2 96(6) 80 80 (6,64) Inflaction Point At x 8 f (8) x3 18(8)2 96(8) 80 48 (8,48) Relative Minimum Questions ? Thanks: For Your Patience !