Answer:
[tex]c=6[/tex]
Step-by-step explanation:
The absolute maximum of a continuous function [tex]f(x)[/tex] is where [tex]f'(x)=0[/tex]. Therefore, we must differentiate the function and then set [tex]x=-5[/tex] and [tex]f'(x)=0[/tex] to determine the value of [tex]c[/tex]:
[tex]f(x)=xe^{-x}+ce^{-x}[/tex]
[tex]f'(x)=-xe^{-x}+e^{-x}-ce^{-x}[/tex]
[tex]0=-(-5)e^{-(-5)}+e^{-(-5)}-ce^{-(-5)}[/tex]
[tex]0=5e^{5}+e^{5}-ce^{5}[/tex]
[tex]0=e^5(5+1-c)[/tex]
[tex]0=6-c[/tex]
[tex]c=6[/tex]
Therefore, when [tex]c=6[/tex], the absolute maximum of the function is [tex]x=-5[/tex].
I've attached a graph to help you visually see this.
