  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: Did you like the article?     (No Ratings Yet) Loading...

Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

Did you like
this resource?

Bravo! Download it in pdf format by simply entering your e-mail!