Question

Let f(x) = xe^-x+ ce^-x, where c is a positive constant. For what positive value of c does f have an absolute

Answer 1

Answer:

$c=6$

Step-by-step explanation:

The absolute maximum of a continuous function $f(x)$ is where $f'(x)=0$. Therefore, we must differentiate the function and then set $x=-5$ and $f'(x)=0$ to determine the value of $c$:

$f(x)=xe^{-x}+ce^{-x}$

$f'(x)=-xe^{-x}+e^{-x}-ce^{-x}$

$0=-(-5)e^{-(-5)}+e^{-(-5)}-ce^{-(-5)}$

$0=5e^{5}+e^{5}-ce^{5}$

$0=e^5(5+1-c)$

$0=6-c$

$c=6$

Therefore, when $c=6$, the absolute maximum of the function is $x=-5$.

I've attached a graph to help you visually see this.