Classify the extreme points of the function, using either the first or second derivative test. Explain why you chose to use the test you did.

Hint

rewrite the function before taking its derivative

Answer

*f* (*x*) = *x*^{2/3}(1 - *x*^{2})

We can rewrite this function as

*f* (*x*) = *x*^{2/3} - *x*^{8/3}.

Then the derivative is

This is undefined when *x* = 0, and zero when

We have critical points at *x* = 0 and .We can't use the second derivative test at *x* = 0 because f'(0) doesn't even exist. The first derivative test it is. Here's the numberline, and the values we need to plug into *f*':

We find

These next ones we'll do with a calculator. If you want to do them by scratch we recommend using *x* = ⅛ instead of *x* = 0.25, since ⅛ has a nice cube root. Remember, all we care about is whether we find positive or negative numbers.

Here's the numberline:

We have a min at *x* = 0, and maxima at *x* = ± ½.