Concavity of Graphs

The definition of the concavity of a graph is introduced along with inflection points. Examples, with detailed solutions, are used to clarify the concept of concavity.

Example 1: Concavity Up

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: Concavity Down

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 (-∞ , + ∞). 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 (-∞ , + ∞) 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.

More References and links

Calculus Tutorials and Problems
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