# How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for # 3x^5 - 5x^3#?

##### 1 Answer

Take the first derivative...

#### Explanation:

Minima and maxima occur at places where the above equation evaluates to zero. Right away, you should be able to see that

But where else?

...which gives you the other points:

To determine whether these might be maxima or minima, you must take the second derivative:

and evaluate at x = -1, 0, and +1.

Finding the regions where the original function is increasing or decreasing requires a little more analysis:

Examine the first derivative equation:

( Eq. 1)

Note that if x < -1, then the terms x+1 and x-1 are both negative, and term

Therefore, in the region

In the region x > 1, the second derivative is obviously positive, so the function is increasing where

In the region

(a similar line of reasoning applies to the region

Always helps to have a graph of the function to serve as a "sanity check".

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

GOOD LUCK!