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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