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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$f(x)=\left\{\begin{array}{clr}\left|2 x^{2}-3 x-7\right| \, \text { if } x \leq-1 \\ {\left[4 x^{2}-1\right]} \text { if } -1 < x < 1 \\ |x+1|+|x-2| \text { if } x \geq 1\end{array}\right.$
$[t]$ denotes the greatest integer $\leq t$, is discontinuous is
$(A)$ $f(x)$ has three real roots if $a > 4$
$(B)$ $f(x)$ has only one real root if $a > 4$
$(C)$ $f(x)$ has three real roots if $a < -4$
$(D)$ $f ( x$ ) has three real roots if $-4 < a < 4$