Intervals of Increase and Decrease

Increasing

If f is differentiable at a:

Decreasing

If f is differentiable at a:

Calculation of the Intervals of Increase and Decrease

Study the intervals of increase and decrease of:

f(x) = x3 − 3x + 2

To determine the intervals of increase and decrease, perform the following steps:

1. Differentiate the function.

f'(x) = 3x2 −3

2. Obtain the roots of the first derivative: f'(x) = 0.

3x2 −3 = 0 x = -1 x = 1

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

4. Take a value from every interval and find the sign they have in the first derivative.

If f'(x) > 0 is increasing.

If f'(x) < 0 is decreasing.

On the interval (−∞, −1), take x = −2, for example.

f'(2) = 3(2)2 −3 > 0

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

f'(0) = 3(0)2 −3 < 0

On the interval (1, ∞), take x = 2, for example.

f'(2) = 3(2)2 −3 > 0

5. Write the intervals of increase and decrease:

Increasing: (−∞, −1) (1, ∞)

Decreasing: (−1,1)