Explanation

(a)
Maximum value of 4 $\sin^{2}{x + 3\cos^{2}{x\ ie,\sin^{2}{x + 3}}}$ is 4 and that of$\ \sin{\frac{x}{2} + \cos{\frac{x}{2}\ is\ \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \sqrt{2}}}$, both attained at $x = \frac{\pi}{2}.$ Hence, the given function has maximum value $4 + \sqrt{2}$