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STD 12 - 6. Application of derivatives — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 6. Application of derivatives4 Marks
MCQ
The maximum value of ${x^{1/x}}$ is
  • A
    ${1 \over e}$
  • ✓
    ${e^{1/e}}$
  • C
    $e$
  • D
    ${1 \over {{e^e}}}$
✓

Answer

Correct option: B.
${e^{1/e}}$
b
(b) $y = {x^{1/x}}$, Taking log ,

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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