Question

Find the absolute extrema of the function. (Round your answer to three decimal places.)
f(x) = xe-x2 on [0,2]

Absolute maximum value
at x =
Absolute minimum value:
at x =

Answer 1

Is that
\[x^e - x^2\]

Answer 2

Take the derivative.
e*x^(e-1) -2x
Set equal to 0 and solve.
e*x^(e-1) = 2x
x^(e-1) = 2/e *x

x^(e-1) / x^1 = 2/e
x^(e-1-1) = 2/e
x^(e-2) = 2/e
(x^(e-2))^(1/e-2) = (2/e)^(1/e-2)
x = (2/e)^(1/e-2) which is about 1.65007509

Check f(x) at that critical point as well as at the endpoints, 0 and 2.

Answer 3

thanks!

Answer 4

My pleasure =D