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. $e^{\pi} \leq \pi^e$

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
$\Rightarrow e^{\frac{1}{e}} > \pi^{\frac{1}{\pi}}$
$\Rightarrow \left(e^{\frac{1}{e}}\right)^{\pi e} > \left(\pi^{\frac{1}{\pi}}\right)^{\pi e}$
$\Rightarrow e^{\pi} > \pi^{e}$