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).
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.
4. Write the intervals:
Convexity: (0, ∞)
Concavity: (−∞, 0)
Example of Intervals of Concavity and Convexity
Convex:
Concave:
