For the function, use the First Derivative Test to determine if each critical point is a minimum, a maximum, or neither.

Answer

We use the quotient rule to find the derivative:

*f *'(*x*) is undefined only at *x* = -3, in which case *f* is also undefined so this is not a critical point. *f *'(*x*) is zero when

*x*^{2} + 6*x* + 5 = 0.

Happily, this quadratic factors as

*x*^{2} + 6*x* + 5 = (*x* + 5)(*x* + 1),

so *f *'(*x*) is zero at *x* = -5 and *x* = -1. Here's the numberline so far:

To figure out whether *f* has a maximum or a minimum at each of these critical points, we need to find the sign of the first derivative so we can fill in the numberline.

The derivative

is negative when exactly one of the quantities (*x* + 5) or (*x* + 1) is negative. This is only possible when

-5 < *x* < -1,in which case (*x* + 1) is negative but (*x* + 5) is positive. So the numberline looks like this:

Since *f *' is positive to the left and negative to the right of *x* = -5, the function *f* has a maximum at *x* = -5. Since *f *' is negative to the left and positive to the right of *x* = -1, the function *f* has a minimum at *x* = -1.