What is the extreme 2xe^{-x^2} ?

The extreme 2xe^{-x^2} is Minimum(-1/(sqrt(2)),-sqrt(2/e)),Maximum(1/(sqrt(2)),sqrt(2/e))

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extreme 2xe^{-x^2}

extreme

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

Solution steps

Find the critical points:

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

Domain of

2 x e^{- x^{2}} : - \infty < x < \infty

Find intervals:

Decreasing

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

Increasing

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

Decreasing

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

Plug the extreme point

x = - \frac{1}{2}

into

2 x e^{- x^{2}} \Rightarrow y = - \frac{2}{e}

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

Plug the extreme point

x = \frac{1}{2}

into

2 x e^{- x^{2}} \Rightarrow y = \frac{2}{e}

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

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

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