The maximum value of $x e^{- x}$ is

e

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\frac{1}{e}

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- e

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- \frac{1}{e}

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Solution

The correct option is B $\frac{1}{e}$
Let $y = x e^{- x}$
Now differentiate it
$\frac{d y}{d x} = e^{- x} - x e^{- x}$ , use product rule
$\Rightarrow \frac{d y}{d x} = e^{- x} (1 - x) = 0$ , for max/min
$\Rightarrow x = 1$
Hence maximum value of given expression occur at $x = 1$
$y_{m a x} = (1) e^{- 1} = \frac{1}{e}$

Note: sign of derivative is changing from positive to negative (increasing to decreasing)
So it will attain maxima at $x = 1$

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