Question

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

• A. $x=e$
• B. $x=1$
• C. $x=\sqrt{e}$
• D. $x=1/2$

Answer 1

Answer: (A)

$x=e$

$Atstationarypoints{f}^{\prime }\left(x\right)=0Nowy=x^{1/x}\therefore logy=\left(\frac{1}{x}\right)\cdot logx\left(\frac{1}{y}\right)\cdot \left(\frac{dy}{dx}\right)=\left(\frac{1/x\cdot x-logx\cdot 1}{x^2}\right)as\left(\frac{dy}{dx}\right)=0\therefore 0=\left(\frac{1-logx}{x^2}\right)orlogx=1\therefore x=e$