What is the secondderivative oftest x*e^{-2*x^2} ?

The secondderivative oftest x*e^{-2*x^2} is Minimum(-1/2 ,-1/(2sqrt(e))),Maximum(1/2 , 1/(2sqrt(e)))

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secondderivative oftest xe^{-2x^2}

second derivative test

x \cdot e^{- 2 \cdot x^{2}}

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

Solution steps

f^{'} (x) = - e^{- 2 x^{2}} (2 x + 1) (2 x - 1)

Find intervals:

Decreasing

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

Increasing

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

Decreasing

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

f^{''} (x) = 16 e^{- 2 x^{2}} x^{3} - 12 e^{- 2 x^{2}} x

Check the sign of

f^{''} (x) = 16 e^{- 2 x^{2}} x^{3} - 12 e^{- 2 x^{2}} x

at each critical point

Check critical point

x = - \frac{1}{2} :

Minimum

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

Check critical point

x = \frac{1}{2} :

Maximum

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

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

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What is the secondderivative oftest x*e^{-2*x^2} ?
The secondderivative oftest x*e^{-2*x^2} is Minimum(-1/2 ,-1/(2sqrt(e))),Maximum(1/2 , 1/(2sqrt(e)))