Maximum value of ( 1 x )^x is - Vidyadip

MCQ

Maximum value of ${\left( {{1 \over x}} \right)^x}$ is

A
${(e)^e}$
✓
${(e)^{1/e}}$
C
${(e)^{ - e}}$
D
${\left( {{1 \over e}} \right)^e}$

✓

Answer

Correct option: B.

${(e)^{1/e}}$

b
(b) $f(x) = {\left( {\frac{1}{x}} \right)^x}$

==> $f'(x) = {\left( {\frac{1}{x}} \right)^x}\left( {\log \frac{1}{x} - 1} \right)$

$f'(x) = 0 \Rightarrow \log \frac{1}{x} = 1 = \log e$

$\Rightarrow \frac{1}{x} = e \Rightarrow x = \frac{1}{e}$

Therefore maximum value of function is ${e^{1/e}}$.

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