$4\sin^2x + 3\cos^2x = 3(\sin^2x + \cos^2x) + \sin^2x = 3 + \sin^2x \le 4$

$\sin\frac{x}{2} + \cos\frac{x}{2} \le \sqrt{2}$ with equality when $\dfrac{x}{2}=\dfrac{\pi}{4}+2n\pi$

When $\sin^2x=1$ and $\sin\frac{x}{2}+\cos\frac{x}{2}=\sqrt{2}$, both maxima occur simultaneously.

Therefore, maximum value $= 4 + \sqrt{2}$

Answer: $\boxed{4+\sqrt{2}}$ ✅