What is the extreme f(x)=8sqrt(x)e^{-x} ?

The extreme f(x)=8sqrt(x)e^{-x} is Minimum(0,0),Maximum(1/2 ,(4sqrt(2))/(sqrt(e)))

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extreme f(x)=8sqrt(x)e^{-x}

extreme points

f (x) = 8 x e^{- x}

Minimum (0, 0), Maximum (\frac{1}{2}, \frac{4 2}{e})

Solution steps

f^{'} (x) = 8 (\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

8 x e^{- x} : 0

Minimum (0, 0)

Plug

x = \frac{1}{2}

into

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

Maximum (\frac{1}{2}, \frac{4 2}{e})

Minimum (0, 0), Maximum (\frac{1}{2}, \frac{4 2}{e})

f (x) = x + sin (x), 0 \leq x \leq 2 π

f (x, y) = 1 - \frac{x ^{2}}{4} - y^{2}

f (x) = x^{3} + 2

f (x) = x^{3} - 1

f (x, y) = x^{4} - \frac{18 x}{12 y ^{2}} - x y^{3} - x^{2}

What is the extreme f(x)=8sqrt(x)e^{-x} ?
The extreme f(x)=8sqrt(x)e^{-x} is Minimum(0,0),Maximum(1/2 ,(4sqrt(2))/(sqrt(e)))