MATH 131-503 Fall 2015 4.3 c Wen Liu 4.3 Derivatives and the Shapes of Curves The Mean Value Theorem If f is a differentiable function on the interval [a, b], then there exists a number c between a and b such that f 0 (c) = f (b) − f (a) b−a or, equivalently, f (b) − f (a) = f 0 (c)(b − a) We can see that this theorem is reasonable by interpreting it geometrically. Increasing/Decreasing Test • If f 0 (x) > 0 on an interval, then f is increasing on that interval. • If f 0 (x) < 0 on an interval, then f is decreasing on that interval. The First Derivative Test Suppose that c is a critical number of a continuous function f . • If f 0 changes from positive to negative at c, then f has a local maximum at c. • If f 0 changes from negative to positive at c, then f has a local minimum at c. • If f 0 does not change sign at c (for example, if f 0 is positive on both sides of c or negative on both sides), then f has no local maximum or minimum at c. Recall: A function (or its graph) is called concave upward on an interval I if f 0 is an increasing function on I. It is called concave downward on I if f 0 is decreasing on I. Concavity Test • If f 00 (x) > 0 for all x ∈ I, then the graph of f is concave upward on I. • If f 00 (x) < 0 for all x ∈ I, then the graph of f is concave downward on I. The Second Derivative Test Suppose f 00 is continuous near c. • If f 0 (c) = 0 and f 00 (c) > 0, then f has a local minimum at c. • If f 0 (c) = 0 and f 00 (c) < 0, then f has a local maximum at c. Page 1 of 3 MATH 131-503 Fall 2015 4.3 c Wen Liu Note: The Second Derivative Test is inconclusive when f 00 (c) = 0. In other words, at such a point there might be a maximum, there might be a minimum, or there might be neither. This test also fails when f 00 (c) does not exist. In such cases the First Derivative Test must be used. Examples: 1. Suppose f 00 is continuous on (−∞, ∞). (a) If f 0 (1) = 0 and f 00 (1) = 1, what can you say about f ? (b) If f 0 (1) = 0 and f 00 (1) = 0, what can you say about f ? 2. Given f (x) = e6x + e−x . Find the local minimum value and the interval on which f is increasing/decreasing and concave up/concave down. Page 2 of 3 MATH 131-503 Fall 2015 4.3 c Wen Liu 3. (p. 276) Discuss the curve f (x) = x2/3 (6 − x)1/3 with respect to concavity, points of inflection, and local maxima and minima. Sketch its graph. Page 3 of 3