The definition of the concavity of graphs is introduced along with inflection points. 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
(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.
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 an 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. An interactive applet on the concavity of graphs quadratic functions is in this site and you can verify the results of this example.