Question

The maximum value of $f\left(x\right)=3\cos {}^{2}x+4\sin {}^{2}x+\cos \frac{x}{2}+\sin \frac{x}{2}$is

• A. $4$
• B. $3+\sqrt{2}$
• C. $4+\sqrt{2}$
• D. none of these

Answer 1

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

Given, $f\left(x\right)=3{\cos }^{2}x+4{\sin }^{2}x+\cos \frac{x}{2}+\sin \frac{x}{2}$
$=3-3{\sin }^{2}x+4{\sin }^{2}x+\sqrt{(\cos \frac{x}{2}+\sin \frac{x}{2}{)}^2}$
$=3+{\sin }^{2}x+\sqrt{1+\sin x}$
we know $-1\le \sin x\le 1$
Hence maximum value of $f\left(x\right)$ is $=4+\sqrt{2}$