Mathematics - Application of Derivative Question with Solution | TestHub

$\begin{array}{rl}& f(x)=3\sqrt{x-2}+\sqrt{4-x}\\ & x-2\ge 0\&4-x\ge 0\\ & \therefore x\in [2,4]\end{array}$
Let $x=2{\sin }^2\theta +4{\cos }^2\theta$ $\begin{array}{rl}& \therefore \mathrm{f}(\mathrm{x})=3\sqrt{2}|\cos \theta |+\sqrt{2}|\sin \theta |\\ & \therefore \sqrt{2}\le 3\sqrt{2}|\cos \theta |+\sqrt{2}|\sin \theta |\le \sqrt{9\times 2+2}\\ & \sqrt{2}\le 3\sqrt{2}|\cos \theta |+\sqrt{2}|\sin \theta |\le \sqrt{20}\\ & \therefore \alpha =\sqrt{2}\beta =\sqrt{20}\\ & {\alpha }^2+2{\beta }^2=2+40=42\end{array}$