# How do you find the turning points of a cubic function?

Mar 30, 2018

Use the first derivative test.

#### Explanation:

Given: How do you find the turning points of a cubic function?

The definition of A turning point that I will use is a point at which the derivative changes sign. According to this definition, turning points are relative maximums or relative minimums.

Use the first derivative test:

First find the first derivative $f ' \left(x\right)$

Set the $f ' \left(x\right) = 0$ to find the critical values.

Then set up intervals that include these critical values.

Select test values of $x$ that are in each interval.

Find out if $f '$(test value $x$) $< 0$ or negative

Find out if $f '$(test value $x$) $> 0$ or positive.

A relative Maximum:
$f ' \left(\text{test value "x) >0, f'("critical value") = 0, f'("test value } x\right) < 0$

A relative Minimum:
$f ' \left(\text{test value "x) <0, f'("critical value") = 0, f'("test value } x\right) > 0$

If you also include turning points as horizontal inflection points, you have two ways to find them:

$f ' \left(\text{test value "x) >0, f'("critical value") = 0, f'("test value } x\right) > 0$

$f ' \left(\text{test value "x) <0, f'("critical value") = 0, f'("test value } x\right) < 0$