Hello everyone,

I’m Prof. Ahmed Hadi and today I’m going to explain the concept of concavity.

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 endeavor. Although modern optimization theory is considerably more advanced, its methods and fundamental ideas clearly show their historical relationship to the calculus. In this lecture, you will review 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.

Let’s start with the definition of the concavity of graphs which is introduced along with inflection points.

The following examples, with detailed solutions, are used to clarify the concept of concavity.

Example 1: Let us consider the graph below. Note that the slope of the tangent line (first derivative) increases. The graph in the figure below is called concave up.

Example 2: The slope of the tangent line (first derivative) decreases in the graph below. We call the graph below concave down.

Definition of Concavity

Let f ' be the first derivative of function f that is differentiable on a given interval I, the graph of f is

(i) concave up on the interval I, if f ' is increasing on I

or

(ii) concave down on the interval I, if f ' is decreasing on I.

The sign of the second derivative informs us when is f ' increasing or decreasing.

Theorem

Let f '' be the second derivative of function f on a given interval I, the graph of f is

 (i) concave up on I if f ''(x) > 0 on the interval I.

(ii) concave down on I if f ''(x) < 0 on the interval I.

Example 3: Determine the values of parameter a for which the graph of function f, defined below, is concave up or down.

f(x) = a x 2 + b x + c

Solution to Example 3:

•   We first find the first and second derivatives of function f.

f '(x) = 2 a x + b

f ''(x) = 2 a

•    We now study the sign of f ''(x) which is equal to 2 a. If a is positive, f ''(x) is positive in the interval (-inf , + inf). According to the theorem above, the graph of f will be concave up for these positive values of a. If a is negative, the graph of f will be concave down on the interval (-inf , + inf) since f ''(x) = 2 a is negative.