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Popular >

extreme f(x)=x^3e^{(4-2x^2)}

Solution

extreme points

f (x) = x^{3} e^{(4 - 2 x^{2})}

Solution

Minimum (- \frac{3}{2}, - \frac{3 3 e ^{\frac{5}{2}}}{8}), Saddle (0, 0), Maximum (\frac{3}{2}, \frac{3 3 e ^{\frac{5}{2}}}{8})

Solution steps

f^{'} (x) = - e^{- 2 (2 + x) (- 2 + x)} x^{2} (2 x + 3) (2 x - 3)

Find intervals:

Decreasing

: - \infty < x < - \frac{3}{2},

Increasing

: - \frac{3}{2} < x < 0,

Increasing

: 0 < x < \frac{3}{2},

Decreasing

: \frac{3}{2} < x < \infty

Plug

x = - \frac{3}{2}

into

x^{3} e^{4 - 2 x^{2}} : - \frac{3 3 e ^{\frac{5}{2}}}{8}

Minimum (- \frac{3}{2}, - \frac{3 3 e ^{\frac{5}{2}}}{8})

Plug

x = 0

into

x^{3} e^{4 - 2 x^{2}} : 0

Saddle (0, 0)

Plug

x = \frac{3}{2}

into

x^{3} e^{4 - 2 x^{2}} : \frac{3 3 e ^{\frac{5}{2}}}{8}

Maximum (\frac{3}{2}, \frac{3 3 e ^{\frac{5}{2}}}{8})

Minimum (- \frac{3}{2}, - \frac{3 3 e ^{\frac{5}{2}}}{8}), Saddle (0, 0), Maximum (\frac{3}{2}, \frac{3 3 e ^{\frac{5}{2}}}{8})

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Frequently Asked Questions (FAQ)

What is the extreme f(x)=x^3e^{(4-2x^2)} ?
The extreme f(x)=x^3e^{(4-2x^2)} is