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

Answer: (A)

$4+\sqrt{2}$

The equation can be expressed as
$\Rightarrow ({\sin }^{2}x+3)+(\cos \frac{x}{2}+\sin \frac{x}{2})$

At $x=\frac{\pi }{2}$ maximum value of

${\sin }^{2}x+3=4$ and

$\cos \frac{x}{2}+\sin \frac{x}{2}=\sqrt{2}$

Therefore the maximum value for $({\sin }^{2}x+3)+(\cos \frac{x}{2}+\sin \frac{x}{2})$ $=4+\sqrt{2}$

Hence, option 'A' is correct.