Answered

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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?

Sagot :

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.

View image goddessboi
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