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*^{4} - 32*x*

Answer

*f* (*x*) = *x*^{4} - 32*x*

The first derivative is

*f*'(*x*) = 4*x*^{3} - 32.

This is zero when *x* = 2, so *x* = 2 is the single critical point of *f*. If we use the first derivative test, we need to evaluate *f*' at two different values of *x*.

Since it's possible to take another derivative, we'll use the second derivative test. The second derivative is

*f*"(*x*) = 12*x*^{2}

which is positive for any *x* ≠ 0. We didn't even need to evaluate *f*" at one value of *x*, which is nice. Since *f*" is positive at *x* = 2, the function *f* is concave up, and therefore has a minimum at *x* = 2.