Question

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

• A. $e^2$
• B. $e^{\left(2-\sqrt{3}\right)}$
• C. $e^4$
• D. $1$

Answer 1

The correct option is C

$e^4$

Explanation for correct option:

Let, $y=e^{\left(2+\sqrt{3}\cos \left(x\right)+\sin \left(x\right)\right)}$

$e^x$ is an increasing function,so $y$ will attain maximum value when $2+\sqrt{3}\cos x+\sin x$ will be maximum.

We know, $-\sqrt{a^2+b^2}\le a\sin x+b\cos x\le \sqrt{a^2+b^2}$

So, maximum value of $2+\sqrt{3}\cos x+\sin x$ is $2+\sqrt{{\left(\sqrt{3}\right)}^2+1^2}=4$

Hence, maximum value of $y$ is $e^4$

Hence the correct option is option C.