Question

The maximum value of $4{\sin }^{2}x+3{\cos }^{2}x+\sin \frac{x}{2}+\cos \frac{x}{2}$ is

• A. $4+\sqrt{2}$
• B. $3+\sqrt{2}$
• C. $9$
• D. $4$

Answer 1

The correct option is A $4+\sqrt{2}$

Maximum value of $4{\sin }^{2}x+3{\cos }^{2}x=4{\sin }^{2}x+3(1-{\sin }^{2}x)={\sin }^{2}x+3$

is $4$ as $0\le {\sin }^{2}x\le 1$

and that of $\sin \frac{x}{2}+\cos \frac{x}{2}$ is $\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\sqrt{2}$

$($ as $-\sqrt{a^2+b^2}\le a\sin x+b\cos x\le \sqrt{a^2+b^2})$

both attained at $x=\frac{\pi }{2}$.

Hence, the given function has maximum value $4+\sqrt{2}$