The function $f (x) = x^{1 / x}$ has stationary point at

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$x = \frac{1}{2}$

$x = 1$

$x = \sqrt{e}$

$x = e$

answer is A.

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Detailed Solution

$y = x^{\frac{1}{x}} \Rightarrow \log y = \frac{1}{x} \log x \Rightarrow \frac{1}{y} \frac{d y}{d x} = \frac{1 - \log x}{x^{2}} ∴ \frac{d y}{d x} = 0 \Rightarrow x = e$

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