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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
Consider the function $f(x) = {e^{ - 2x}}$ $sin\, 2x$ over the interval $\left( {0,{\pi \over 2}} \right)$. A real number $c \in \left( {0,{\pi \over 2}} \right)\,,$ as guaranteed by Rolle’s theorem, such that $f'\,(c) = 0$ is
  • ✓
    $\pi /8$
  • B
    $\pi /6$
  • C
    $\pi /4$
  • D
    $\pi /3$
✓

Answer

Correct option: A.
$\pi /8$
a
(a) $f(x) = {e^{ - 2x}}\sin 2x$ ==> $f'(x) = 2{e^{ - 2x}}(\cos 2x - \sin 2x)$

Now,   $f'(c) = 0$

==>$\cos 2c - \sin 2c = 0$==>$\tan 2c = 1$==>$c = \frac{\pi }{8}$.

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