Question

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

Answer 1

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'(x)`

Set the `f'(x) = 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'("test value "x) >0, f'("critical value") = 0, f'("test value "x) < 0`

A relative Minimum:
`f'("test value "x) <0, f'("critical value") = 0, f'("test value "x) > 0`

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

`f'("test value "x) >0, f'("critical value") = 0, f'("test value "x) > 0`

`f'("test value "x) <0, f'("critical value") = 0, f'("test value "x) < 0`