What is the extreme f(x)=4sqrt(x)e^{-x} ?

The extreme f(x)=4sqrt(x)e^{-x} is Minimum(0,0),Maximum(1/2 ,(2sqrt(2))/(sqrt(e)))

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extreme f(x)=4sqrt(x)e^{-x}

extreme points

f (x) = 4 x e^{- x}

Minimum (0, 0), Maximum (\frac{1}{2}, \frac{2 2}{e})

Solution steps

f^{'} (x) = 4 (\frac{e ^{- x}}{2 x} - e^{- x} x)

Find intervals:

Increasing

: 0 < x < \frac{1}{2},

Decreasing

: \frac{1}{2} < x < \infty

Plug

x = 0

into

4 x e^{- x} : 0

Minimum (0, 0)

Plug

x = \frac{1}{2}

into

4 x e^{- x} : \frac{2 2}{e}

Maximum (\frac{1}{2}, \frac{2 2}{e})

Minimum (0, 0), Maximum (\frac{1}{2}, \frac{2 2}{e})

f (x) = \frac{e ^{x}}{x ^{10}}

f (x, y) = (x^{2} + y^{2}) (e^{y^{2} - x^{2}})

f (x, y) = e^{x + y}

f (x) = x + x^{3}

f (x) = 3 x^{3} - 4 x

What is the extreme f(x)=4sqrt(x)e^{-x} ?
The extreme f(x)=4sqrt(x)e^{-x} is Minimum(0,0),Maximum(1/2 ,(2sqrt(2))/(sqrt(e)))