# How do you find the local max and min for #f(x) = x^3 - 27x#?

##### 1 Answer

Maximum at

#### Explanation:

Find the **critical values** of

A critical value

Find

#f(x)=x^3-27x#

#f'(x)=3x^2-27#

#3c^2-27=0#

#3c^2=27#

#c^2=9#

#c=+-3#

We know that two critical values, at which maxima or minima could occur, are

We can use either or the first or second derivative test to determine if these are minima or maxima.

**First Derivative Test**

Examine the change in the function surrounding the critical values.

#f'(-4)=21larr"increasing"#

#f'(-3)=0#

#f'(-2)=-15larr"decreasing"#

Since the derivative changes from increasing to decreasing when

#f'(2)=-15larr"decreasing"#

#f'(3)=0#

#f'(4)=21larr"increasing"#

Since the derivative changes from decreasing to increasing when

**Second Derivative Test**

Examine the concavity at each point to determine whether a minimum or maximum should occur.

First, find

#f'(x)=3x^2-27#

#f''(x)=6x#

Now, find the concavity at each of the critical values.

#f''(-3)=-18#

Since *relative maximum* when

#f''(3)=18#

Since *relative minimum* when