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6. Application of derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths6. Application of derivatives1 Mark
MCQ
The function $f(x) = {x^{1/x}}$ is
  • A
    Increasing in $(1,\,\,\infty )$
  • B
    Decreasing in $(1,\,\,\infty )$
  • ✓
    Increasing in $(1,\,e),$ decreasing in $(e,\infty )$
  • D
    Decreasing in $(1,\,e),$ increasing in$(e,\infty )$
✓

Answer

Correct option: C.
Increasing in $(1,\,e),$ decreasing in $(e,\infty )$
c
(c) Let $y = {x^{1/x}}$==> $\log y = \frac{1}{x}\log x$

==>$\frac{1}{y}\frac{{dy}}{{dx}} = \frac{1}{{{x^2}}} - \frac{{\log x}}{{{x^2}}} = \frac{{1 - \log x}}{{{x^2}}}$

==>$\frac{{dy}}{{dx}} = {x^{1/x}}\left( {\frac{{1 - \log x}}{{{x^2}}}} \right)$

Now, ${x^{1/x}} > 0$for all x and $\frac{{1 - \log x}}{{{x^2}}} > 0$ in $ (1, e) $ and 

$\frac{{1 - \log x}}{{{x^2}}} < 0$ in $(e,\infty )$

$\therefore$ $f(x)$ is increasing in $ (1, e) $ and decreasing in $(e,\,\infty ).$

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