# How do use the first derivative test to determine the local extrema #F(x) = -2x^3 - 9x^2 + 24x + 40#?

##### 1 Answer

#### Answer:

Determine the critical points of the function and check to see if the first derivative changes sign around these points.

#### Explanation:

The **first derivative test** allows you to determine whether or not a *critical point* of a function is also a *local minimum* or a *local maximum*.

This can be done by checking to see if the first derivative of the function changes signs around these critical points.

The idea is that a critical point is a **local minimum** if the function goes from **decreasing** to **increasing**, i.e. if *negative* to positive*, around that point.

Likewise, a critical point is a **local maximum** if the function goes from **increasing** to **decreasing**, i.e. if *positive* to *negative*, around that point.

So, start by determining the first derivative for your function

This is equivalent to

To determine the *critical points* of the function, make

Since you have no domain restrictions for this function, both solutions will be *critical points*.

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

Select a value from each of these intervals and use it to determine the sign of

#(-oo,-4)#

#(-4,1)#

#(1, oo)#

So, the first derivative changes sign **twice**. It first goes from **negative** to **positive** around **positive** to **negative** around

This means that the function switches from *decreaing* to *increasing* around **local minimum**.

Then the function switches from *increasing* to *decreasing* around **local maximum**.

To get the actual coordinates of the local extrema, evaluate the function

and

Therefore, your function has

*local minimum*

*local maximum*