# 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) = x^{3} − 3x + 2

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

1. Differentiate the function.

f'(x) = 3x^{2} −3

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

3x^{2} −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)