Concavity and the second derivative

One of the most important applications of differential calculus is to find extreme function values. The calculus methods for finding the maximum and minimum values of a function are the basic tools of optimization theory, a very active branch of mathematical research applied to nearly all fields of practical endeavour. Although modern optimization theory is considerably more advanced, its methods and fundamental ideas clearly show their historical relationship to the calculus. Here we are going to discuss how the second derivative of a function is related to the shape of its graph and how that information can be used to classify relative extreme values.

Concavity

The Second Derivative Test provides a means of classifying relative extreme values by using the sign of the second derivative at the critical number. To appreciate this test, it is first necessary to understand the concept of concavity.

The graph of a function f is concave upward at the point (c f(c)) if f`(c) exists and if for all x in some open interval containing c , the point (x f(x)) on the graph of f lies above the corresponding point on the graph of the tangent line to f at c . This is expressed by the inequality f(x) [f(c)+f (c)(x-c)] for all x in some open interval containing c . Imagine holding a ruler along the tangent line through the point (c f(c)) if the ruler supports the graph of f near (c f(c)) , then the graph of the function is concave upward.

The graph of a function f is concave downward at the point (c f(c)) if f (c) exists and if for all x in some open interval containing c , the point (x f(x)) on the graph of f lies below the corresponding point on the graph of the tangent line to f at c . This is expressed by the inequality f(x) [f(c)+f (c)(x-c)] for all x in some open interval containing c . In this situation the graph of f supports the ruler.

Concavity theorem

Concavity theorem is the important result that relates the concavity of the graph of a function to its derivatives and it is as follows:

If the function f is twice differentiable at x = c, then the graph of f is concave upward at (c, f(c)) if f''(c) > 0 and concave downward if f''(c) < 0 .

Points on the graph of f where the concavity changes from up-to-down or down-to-up are called inflection points of the graph.

Second derivative test

The Second Derivative Test relates the concepts of critical points, extreme values, and concavity to give a very useful tool for determining whether a critical point on the graph of a function is a relative minimum or maximum.

Suppose that c is a critical point at which f'(c) = 0, that f'(x) exists in a neighbourhood of c, and that f''(c) exists. Then f has a relative maximum value at c if f''(c) < 0 and a relative minimum value at c if f''(c) > 0. If f''(c) = 0, then the test is not informative.