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

1 Answer
marfre
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'(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

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