Problem 7

$\begin{aligned} y & = {\color{#3D99F6}4\sin^2 x + 3\cos^2 x} + \sin \frac x2 + \cos \frac x2 & \small \color{#3D99F6} \text{Since }\sin^2 \theta + \cos^2 \theta = 1 \\ & = {\color{#3D99F6} \sin^2 x + 3} + \sqrt 2 \left(\color{#D61F06}\frac 1{\sqrt 2}\sin \frac x2 + \frac 1{\sqrt 2}\cos \frac x2 \right) & \small \color{#D61F06} \text{and }\sin (A+B) = \sin A \cos B + \sin B \cos A \\ & = \sin^2 x + 3 + \sqrt 2 \color{#D61F06}\sin \left(\frac x2 + \frac \pi 4\right) \end{aligned}$

Note that $\max (\sin x) = \max \left(\sin \left(\dfrac x2 + \dfrac \pi 4\right)\right) = 1$ , when $x = \dfrac \pi 2$

$\implies \max (y) = 1 + 3 + \sqrt 2 \approx \boxed{5.414}$