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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 $f(x) = {x^{ - x}},\,(x\, \in \,R)$ attains a maximum value at $x =$
  • A
    $x =  2$
  • B
    $x = 3$
  • ✓
    $x = 1/e $
  • D
    $x = 1 $
✓

Answer

Correct option: C.
$x = 1/e $
c
(c) $f(x) = y = {x^{ - x}}$ ==> $\log y = - \,x\log x$

Differentiating w.r.t.  $x$ , $\frac{1}{y}.\frac{{dy}}{{dx}} = - \left[ {x.\frac{1}{x} + \log x} \right]$

==> $\frac{1}{y}.\frac{{dy}}{{dx}} = - [1 + \log x]$ ==> $\frac{{dy}}{{dx}} = - {x^{ - x}}[1 + \log x]$

==> $\frac{{dy}}{{dx}} = {x^{ - x}}\left[ {\log \frac{1}{x} - 1} \right]$

Put $\frac{{dy}}{{dx}} = 0$ ==> ${\log _e}\frac{1}{x} = {\log _e}e$

==> $\frac{1}{x} = e \Rightarrow x = \frac{1}{e}$.

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