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



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