Question

5.11 solving optimization problems
test prep

1. let $f(x) = xe^{-x} + ce^{-x}$, where $c$ is a positive constant. for what positive value of $c$ does $f$ have an absolute maximum at $x = -5$?

Explanation:

Step1: Simplify f(x)

$f(x) = e^{-x}(x + c)$

Step2: Compute derivative f’(x)

Using product rule: $f'(x) = -e^{-x}(x + c) + e^{-x}(1) = e^{-x}(1 - x - c)$

Step3: Set f’(-5)=0

At $x=-5$, $f'(-5)=e^{5}(1 - (-5) - c)=e^5(6 - c)=0$. Since $e^5≠0$, $6 - c=0$

Step4: Solve for c

$c=6$

Answer:

6