Question

Find 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 $4+\sqrt{2}$

Max. value of $4{\sin }^{2}x+3{\cos }^{2}x={\sin }^{2}x+3$ is $4$ hold at $x=\frac{x}{2}s$ and max. value of $\sin \frac{x}{2}+\cos \frac{x}{2}$ is $\sqrt{2}$ at $x=\frac{\pi }{2}$

Since the maximum value occurs at the same point for the two expressions, the maximum value of $4{\sin }^{2}x+3{\cos }^{2}x+\sin \frac{x}{2}+\cos \frac{x}{2}$ is $4+\sqrt{2}$