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.

*f* (*x*) = (*x* - 1)^{9}

Answer

The first derivative is

This one is nasty. We know that *f*' is only undefined for *x* = , in which case *f* is also undefined, so we don't find any critical points that way. To find when *f*' is zero we need to solve the equation

*x*^{4} - 10*x*^{2} - 3 = 0.

We can use the quadratic formula to solve for *x*^{2}:

Since is negative and *x*^{2} can't be negative, we know

Therefore

Now it's time to choose the first or second derivative test. If we use the second derivative test we need to find the second derivative, which requires the quotient rule again.

Instead, use the first derivative test. We have this numberline:

The numbers 0, ± 4 seem like nice ones to plug into the first derivative.

Since the first derivative contains only even powers of *x*, we know

f'(4) = f'(-4)

so we only need to evaluate one of these.

Now the numberline looks like this:

and we know that there is a maximum at

and a minimum at