Question

Maximum value of the expression $2\sin x+4\cos x+3$ is :

• A. $2\sqrt{5}+3$
• B. $2\sqrt{5}-3$
• C. $\sqrt{5}+3$
• D. None of these

Answer 1

Answer: (A)

$2\sqrt{5}+3$

Let $f\left(x\right)=2\sin x+4\cos x+3$

$f\left(x\right)$ can be written as $\sqrt{20}\times \frac{2\sin x+4\cos x}{\sqrt{20}}+3$

$\Rightarrow f\left(x\right)=\sqrt{20}\times (\cos \alpha \sin x+\sin \alpha \cos x)+3$ where $\cos \alpha =\frac{2}{\sqrt{20}}$

$\Rightarrow f\left(x\right)=\sqrt{20}\times \sin (\alpha +x)+3$

Since the maximum value of $\sin (\alpha +x)$ can be $1$, the maximum value of $f\left(x\right)$ becomes $\sqrt{20}+3$, i.e. $2\sqrt{5}+3$