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

*f*'(*x*) = (*x* + 5)(*x* + 1)

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.