Question

Find Maxima and Minima $y=x^{\frac{1}{x}}$

Answer 1

$y=x^{1/x}$

$\log y=\frac{1}{x}\log x$.

$\frac{y^{\prime }}{y}=\frac{1}{x}\frac{1}{x}+\frac{-1}{x^2}\log x$

$\frac{d}{dx}\left(x^{y_n}\right)=(1-\log x)x^{y_n-2}$

$\because y=0,(1-\log x)x^{y_n-2}=0$

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

$x=e,o$

If $x=e\Rightarrow y=e^{ye}$

If $x=o,y=o=A$, so $(e,e^{1/e})$ is the solution