What is the extreme f(x)=10x^{3/4}e^{-x} ?

The extreme f(x)=10x^{3/4}e^{-x} is Minimum(0,0),Maximum(3/4 ,(5*3^{3/4})/(2^{1/2)e^{3/4}})

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extreme f(x)=10x^{3/4}e^{-x}

extreme points

f (x) = 10 x^{\frac{3}{4}} e^{- x}

Minimum (0, 0), Maximum (\frac{3}{4}, \frac{5 \cdot 3 ^{\frac{3}{4}}}{2 ^{\frac{1}{2}} e ^{\frac{3}{4}}})

Solution steps

Find the critical points:

x = 0, x = \frac{3}{4}

Domain of

10 x^{\frac{3}{4}} e^{- x} : x \geq 0

Find intervals:

Increasing

: 0 < x < \frac{3}{4},

Decreasing

: \frac{3}{4} < x < \infty

Plug the extreme point

x = 0

into

10 x^{\frac{3}{4}} e^{- x} \Rightarrow y = 0

Minimum (0, 0)

Plug the extreme point

x = \frac{3}{4}

into

10 x^{\frac{3}{4}} e^{- x} \Rightarrow y = \frac{5 3 ^{\frac{3}{4}}}{2 ^{\frac{1}{2}} e ^{\frac{3}{4}}}

Maximum (\frac{3}{4}, \frac{5 \cdot 3 ^{\frac{3}{4}}}{2 ^{\frac{1}{2}} e ^{\frac{3}{4}}})

Minimum (0, 0), Maximum (\frac{3}{4}, \frac{5 \cdot 3 ^{\frac{3}{4}}}{2 ^{\frac{1}{2}} e ^{\frac{3}{4}}})

f (x) = 2 x - 4 y + 6 z = 56

y = 1 - x^{2} - z^{2}

20 x^{2} - 200 x + 3

y = \frac{x ^{4}}{1 + x ^{3}}

\frac{200}{x + 4 + 2 x}

What is the extreme f(x)=10x^{3/4}e^{-x} ?
The extreme f(x)=10x^{3/4}e^{-x} is Minimum(0,0),Maximum(3/4 ,(5*3^{3/4})/(2^{1/2)e^{3/4}})