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Applications of Derivative — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplications of Derivative1 Mark
MCQ
Maximum value of $=\left(\frac{1}{x}\right)^x$:
  • ✓
    $e^{1 / e}$
  • B
    $e$
  • C
    $\left(\frac{1}{e}\right)^{1 / e}$
  • D
    $e^{\varepsilon}$
✓

Answer

Correct option: A.
$e^{1 / e}$
(A)$y=\left(\frac{1}{x}\right)^x$
$\therefore \quad \log y=\log \left(\frac{1}{x}\right)^x=-x \log x$
$\therefore \quad \frac{1}{y} \frac{d y}{d x}=-x \frac{1}{x}-\log x=-(1+\log x)$
$\Rightarrow \quad \frac{d y}{d x}=-y(1+\log x)$
now $\frac{d y}{d x}=0 \Rightarrow 1+\log x=0$
$\Rightarrow \log x=-1 \Rightarrow x=e^{-1}=\frac{1}{e}$
hence maximum value $=\left(\frac{1}{1 / e}\right)^{1 / e}=e^{1 / e}$

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