Note: One of the nicest things to do is to graph the function, if you have that option available. A simple look at the graph will tell you that the answer is e). Failing that possibility:

by the product rule.

We need to find where this is zero.

We only need to numerator to be 0, so:

Divide both sides by

(which is never 0):

Now solve for x:

Thus x = -1. (Note: this value of x does NOT make the denominator 0.)

Now, the point (-1, f(-1)) is merely a critical point. It may be a max or a min and it may only be local. It may also be an inflection point. We need to check all these possibilities.

First comment: The function f(x) has no vertical asymptotes.

A reasonable check on whether the critical point is a local or absolute maximum (in the absence of vertical asymptotes) is to take the limit of the function as x goes to positive and negative infinity to see if it "blows up" or "blows down."

Since your function goes to infinity as x approaches to negative infinity, your critical point could only possibly be a local max at best. (As it happens, x = -1 is actually an inflection point, not a local max or min.)

Thus the function has no absolute maximum.

-Dan