# How do use the first derivative test to determine the local extrema #f(x) = 3x^5 - 20x^3#?

##### 1 Answer

#### Answer:

You determine the crititcal points of the function and check to see if any of these points are local minima or local maxima.

#### Explanation:

Basically, the **first derivative test** allows you to look for a function's *critical points* and check to see whether or not these points are *local extrema* by looking if the sign of the first derivative **changes** around them.

So, the first thing you need to do is determine the first derivative of

To find the function's *critical points*, make the first derivative equal to zero and solve for

This equation has three solutions,

Since your function has no domain restrictions, all these points will be *critical points*.

Now, in order for the points to be local extrema, the function must go from **increasing**, i.e. a *positive* **decreasing**, i.e. a *negative*

Since you have **3 critical points**, you're going to look at **4** intervals. Select a value from each itnerval to determine the sign of the first derivative on that interval

#(-oo, -2)#

#(-2, 0)#

#(0, 2)#

#(2, 0)#

So, you know that the first derivative **changes sign**, i.e. it goes from positive to negative, around **local maximum**.

On the other hand, the first derivative **does not** change sign around point

Notice that the first derivative **changes sign** again around point *local minimum*.

To get the actual points that match these critical points, evaluate the original functions in

and

Therefore, the function **local minimum** at **local maximum** at