`f(x) = (x^2)e^(-x)` (a) Use a graph of `f` to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of `x` at which `f` increases most rapidly. Then find...

`f(x) = (x^2)e^(-x)` (a) Use a graph of `f` to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of `x` at which `f` increases most rapidly. Then find the exact value.

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gsarora17 | (Level 2) Associate Educator

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a)  Graph is attached . Minimum value of function is 0 at x=0

`f(x)=x^2e^-x`

`f'(x)=x^2d/dxe^-x+e^-xd/dxx^2`

`f'(x)=x^2(-1e^-x)+e^-x(2x)`

`f'(x)=e^-x(-x^2+2x)`

`f'(x)=-e^-x*x(x-2)`

So the critical numbers can be evaluated for f'(x)=0

Critical numbers are x=0 , x=2

f(0)=0

So the function has Minimum=0 at x=0

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