Question

Given that $f(x)=x^{1/x},x>0$ has the maximum value at $x=e,$ then

• A. $e^{\pi }>{\pi }^e$
• B. $e^{\pi }<{\pi }^e$
• C. $e^{\pi }={\pi }^e$
• D. $e^{\pi }\le {\pi }^e$

Answer 1

Answer: (A)

$f(x)-x^{1/x},x>0$. Since $x=e$ is $a$ pt. of maxima.
$\therefore f\left(e\right)>f\left(x\right)$ for all $x>0$
$\Rightarrow f\left(e\right)>f\left(\pi \right)$ in particular
$\Rightarrow e^{\frac{1}{e}}>{\pi }^{\frac{1}{\pi }}$
$\Rightarrow {\left(e^{\frac{1}{e}}\right)}^{\pi e}>{\left({\pi }^{\frac{1}{\pi }}\right)}^{\pi e}$
$\Rightarrow e^{\pi }>{\pi }^e$