Concavity of Graphs




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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.

concave up example 1
Figure 1

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

concave down example 2
Figure 2

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 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.
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Updated: 27 November 2007 (A Dendane)