The function f ( x ) = x x has a stationary point at:

Question

The function $f (x) = x^{x}$ has a stationary point at:

Smart Achievers · Accepted Answer

(2)

Ans. $x = \frac{1}{e}$

Explanation:

We have, $f (x) = x^{x}$

Let $y = x^{x}$

and $\log y = x \log x$

$∴$ $\frac{1}{y} \cdot \frac{d y}{d x} = x \cdot \frac{1}{x} + \log x \cdot 1$

$\Rightarrow$ $\frac{d y}{d x} = (1 + \log x) x^{x}$

$∴$ $\frac{d y}{d x} = 0$

$\Rightarrow$ $(1 + \log x) x^{x} = 0$

$\Rightarrow$ $\log x = - 1$

$\Rightarrow$ $\log x = \log e^{- 1}$

$\Rightarrow$ $x = e^{- 1}$

$\Rightarrow$ $x = \frac{1}{e}$

Hence, $f (x)$ has a stationary point at $x = \frac{1}{e}$ .

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