Question

$f(x)=x^x$ has stationary point at

• A. $x=e$
• B. $x=\frac{1}{e}$
• C. $x=1$
• D. $x=\sqrt{e}$

Answer 1

Answer: (B)

Given $f(x)=y=x^x$
$\Rightarrow \log y=x\log x$
Differentiating, we get
$\frac{1}{y}.\frac{dy}{dx}=\frac{x}{x}+\log x$
$\Rightarrow \frac{dy}{dx}=y\left[1+\log x\right]$
At the stationary point, $\frac{dy}{dx}=0$
$\Rightarrow x^x\left[1+\log x\right]=0$
$\Rightarrow {\log }_{e}x=-1$
$\Rightarrow x=\frac{1}{e}$