Concave and Convex Functions

Concave and Convex Functions

Concave and Convex Functions

Intervals of Concavity and Convexity

Study the intervals of concavity and convexity of the following function:

f(x) = x3 − 3x + 2

To study the concavity and convexity, perform the following steps:

1. Find the second derivative and calculate its roots.

f''(x) = 6x 6x = 0x = 0.

2. Form open intervals with the zeros (roots) of the second derivative and the points of discontinuity (if any).

 Interval

3. Choose a value in each interval and determine the sign that is in the second derivative.

If f''(x) > 0 it is convex.

If f''(x) < 0 it is concave.

For the interval (− ∞, 0), take x = −1, for example.

f''(−1) = 6(−1) < 0 Concave.

For the interval (0, ∞), take x = 1, for example.

f''(1) = 6 (1) > 0 Convex.

Concave and Convex Functions

4. Write the intervals:

Convexity: (0, ∞)

Concavity: (−∞, 0)

Example of Intervals of Concavity and Convexity

Function

Domain

Concave and Convex Function Operations

Concave and Convex Function Operations

Concave and Convex Function Operations

Concave and Convex Function Operations

Concave and Convex Function Solution

Convex:

Concave and Convex Function Solution

Concave:

Concave and Convex Function Solution