Maximum value of (\left( \frac{1}{x} \right)^{x}) is | MATH - MAXIMA AND MINIMA | SolveeIt

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

$(e)^{e}$

$(e)^{1/e}$

$(e)^{- e}$

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

Answer

$(e)^{1/e}$

Explanation

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

$f^{'}(x) = 0$ ⇒ $\log\frac{1}{x} = 1 = \log e$ ⇒ $\frac{1}{x} = e$ ⇒ $x = \frac{1}{e}$ .

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

Question: Maximum value of \(\left( \frac{1}{x} \right)^{x}\) is...