# How do you find all relative extrema of the function #f(x)= -x^3 -6x^2-9x-2#?

##### 1 Answer

#### Answer:

Use the first derivative test and check for sign changes of

#### Explanation:

For a given function, *relative extrema*, or *local maxima and minima*, can be determined by using the **first derivative test**, which allows you to check for any sign changes of *critical points*.

For a *critical point* to be *local extrema*, the function must go from **increasing**, i.e. *positive* **decreasing**, i.e. *negative*

So, start by determining the first derivative of

To determine the function's critical points, make

This is equivalent to

Since no domain restrictions are given for your function, both solutions will be *critical points*.

Now check to see if the first derivative changes sign around these points. Since you're dealing with two critical points, you're going to have to look at **3 intervals**.

Select a value fro meach of these intervals and note the sign of

#(-oo,-3)#

#(-3,-1)#

#(-1, oo)#

The first derivative changes sign **twice**. It goes from being *negative* to being *positive* around **local minimum**.

On the other hand, it goes from being *positive* to being *negative* around point **local maximum**.

This is equivalent to having a function that goes from **decreasing** to **increasing** (think of a valley) around point **increasing** to **decreasing** (think of a hill) around point

To get the actual points at which the function has the local minium and maximum, evaluate

and

Therefore, the function

*local minimum*

*local maximum*

graph{-x^3 - 6x^2 - 9x - 2 [-10, 10, -5, 5]}