If the function f(x)= (1 x )^ 2 x ; x>0 attains the maximum value… - Vidyadip

MCQ

If the function $f(x)=\left(\frac{1}{x}\right)^{2 x} ; x>0$ attains the maximum value at $x =\frac{1}{ e }$ then :

A
$e ^\pi<\pi^{ e }$
B
$e ^{2 \pi}<(2 \pi)^e$
C
$e ^\pi>\pi^{ e }$
D
$(2 e )^\pi>\pi^{(2 e )}$

✓

Answer

Let $y=\left(\frac{1}{x}\right)^{2 x}$
$\ell n y=2 x \ell\ n\left(\frac{1}{x}\right)$
$\ell n y=-2 x \ell \ n x$
$\frac{1}{y} \frac{d y}{d x}=-2(1+\ell\ n x)$
for $x >\frac{1}{ e } f ^{ n }$ is decreasing
so $, e<\pi$
$\left(\frac{1}{e}\right)^{2 e}>\left(\frac{1}{\pi}\right)^{2 \pi}$
$e^\pi>\pi^e$

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