This function has an absolute maximum value equal to _________ which is attained at x = _________?

Question

Consider the function f(x)=xe^−2x, 0 ≤ x ≤ 2.

This function has an absolute maximum value equal to _________ which is attained at x = _________?

Answer 1

Recall that absolute maxima can occur in three possible places:

1) At the beginning of a closed interval

2) At the end of a closed interval

3) Where the derivative changes from positive to negative (where the function goes from increasing to decreasing)

So, all we have to do to solve this problem is evaluate the function at 0, at 2, and where the derivative changes from positive to negative.

Of course, finding the derivative of xe^-2x requires the product rule and the chain rule. After applying these rules, the derivative should be e^-2x + -2xe^-2x. Now, we can use this derivative to find the critical points.

e^-2x + -2xe^-2x = 0 implies that e^-2x = 2xe^-2x. Since, e^-2x is never zero in the real numbers, we can cancel it out on both sides leaving us with 1 = 2x. Thus, the derivative is zero when x = 1/2.

Now all we have to do is see which is largest: f(0), f(1/2), or f(2).

f(0) = 0.

f(1/2) = (1/2)*e^-1.

f(2) = 2*e^-4.

A rough estimate (or the use of a calculator) will show you that 1/2*e^-1 is the largest. So, the absolute maximum is attained at x = 1/2 and is 1/2*e^-1.