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Popular >

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

Solution

extreme points

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

Solution

Minimum (- \frac{2 + 10}{2}, - \frac{e ^{2 + 10} ( 10 - 1 )}{2}), Maximum (\frac{- 2 + 10}{2}, \frac{e ^{2 - 10} ( 1 + 10 )}{2})

Solution steps

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

Find intervals:

Decreasing

: - \infty < x < - \frac{2 + 10}{2},

Increasing

: - \frac{2 + 10}{2} < x < \frac{5}{2} - 1,

Decreasing

: \frac{5}{2} - 1 < x < \infty

Plug

x = - \frac{2 + 10}{2}

into

x (x + 3) e^{- 2 x} : - \frac{e ^{2 + 10} ( 10 - 1 )}{2}

Minimum (- \frac{2 + 10}{2}, - \frac{e ^{2 + 10} ( 10 - 1 )}{2})

Plug

x = \frac{- 2 + 10}{2}

into

x (x + 3) e^{- 2 x} : \frac{e ^{2 - 10} ( 1 + 10 )}{2}

Maximum (\frac{- 2 + 10}{2}, \frac{e ^{2 - 10} ( 1 + 10 )}{2})

Minimum (- \frac{2 + 10}{2}, - \frac{e ^{2 + 10} ( 10 - 1 )}{2}), Maximum (\frac{- 2 + 10}{2}, \frac{e ^{2 - 10} ( 1 + 10 )}{2})

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Frequently Asked Questions (FAQ)

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