MCQ
The maximum value of ${x^{1/x}}$ is
- A${1 \over e}$
- ✓${e^{1/e}}$
- C$e$
- D${1 \over {{e^e}}}$
we have $\log y = \frac{1}{x}\log x$
Differentiate both sides w.r.t. $x $
$\frac{1}{y}\frac{{dy}}{{dx}} = \frac{1}{{{x^2}}} - \frac{{\log x}}{{{x^2}}}$
==> $\frac{{dy}}{{dx}} = \frac{1}{{{x^2}}}(1 - \log x){x^{1/x}}$
For maximum, $\frac{{dy}}{{dx}} = 0$ ==> $x = e$;
$\therefore$ ${y_{\max }} = {e^{1/e}}$.
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$2 x+y-z=5$
$2 x-5 y+\lambda z=\mu$
$x+2 y-5 z=7$
has infinitely many solutions, then $(\lambda+\mu)^2+(\lambda-\mu)^2$ is equal to