Question 4
Find the largest value of
$$\frac1{\sqrt x}+\frac2{\sqrt y}+\frac3{\sqrt z}$$
for positive numbers $x,y,z$ satisfying
$$\begin{cases}\dfrac1{\sqrt2}\le z\le\min\{x\sqrt2,y\sqrt3\},\\[3pt]x+z\sqrt3\ge\sqrt6,\\[3pt]y\sqrt3+z\sqrt{10}\ge2\sqrt5.\end{cases}$$
$$\frac1{\sqrt x}+\frac2{\sqrt y}+\frac3{\sqrt z}$$
for positive numbers $x,y,z$ satisfying
$$\begin{cases}\dfrac1{\sqrt2}\le z\le\min\{x\sqrt2,y\sqrt3\},\\[3pt]x+z\sqrt3\ge\sqrt6,\\[3pt]y\sqrt3+z\sqrt{10}\ge2\sqrt5.\end{cases}$$