Find the maximum value?

Question

Find the maximum value of the function f(x)=xe^-x. 

 

Answer: 1/e

 

Please show all your work. 

Answer 1

y = xe^-x

 

To find the maximum value, take the derivative of y wrt x, set it to zero, and solve for x.  Note that y is the product of two functions, so you will need to use the product rule (if y = g(x)h(x), then y' = g'(x)h(x)+g(x)h'(x) ).  You'll also need to use the chain rule to take the derivative of e^-x.

 

y' = dy/dx = d(x)/dx*(e^-x)+x*d(e^-x)/dx = e^-x - xe^-x = e^-x(1 - x)= 0 when x = 1

 

Plug x=1 back into the original equation (y = xe^-x ) to get the value of the function at x=1.