The maximum value of e^ (2 + 3 x + x) is - Vidyadip

MCQ

The maximum value of $e^{(2 + \sqrt 3 \cos x + \sin x)}$ is

A
$e^{(2)}$
B
$e^{(2 - \sqrt 3 )}$
✓
$e^{(4)}$
D
$1$

✓

Answer

Correct option: C.

$e^{(4)}$

c
(c) Let $y = \exp (2 + \sqrt 3 \cos x + \sin x)$

==> $y' = \exp (2 + \sqrt 3 \cos x + \sin x)\,( - \sqrt 3 \sin x + \cos x)$

Now $y' = 0$ ==> $ - \sqrt 3 \sin x + \cos x = 0$

==> $\sin \left( {x - \frac{\pi }{6}} \right) = 0$ ==> $x = \frac{\pi }{6}$

Now $y''$ is $-ve$ at $x = \frac{\pi }{6}$

Maximum value of

$y = \exp \,\left( {2 + \sqrt 3 \left( {\frac{{\sqrt 3 }}{2}} \right) + \frac{1}{2}} \right)$

$= \exp (4)$.

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