Cauchy-Schwarz inequality with roots

Since $x>0$ , $y>0$ and $z>0$ , by the Cauchy-Schwarz inequality , we have

$\begin{aligned} (1^2+5 ^2+7 ^2)(x+y+z) & \geq \left(\sqrt{x}+5\sqrt{y}+7\sqrt{z}\right)^2 \\ 75 ^2 & \geq \left(\sqrt{x}+5\sqrt{y}+7\sqrt{z}\right)^2 \\ 0 < \sqrt{x}+5\sqrt{y}+7\sqrt{z} & \leq 75, \end{aligned}$ where equality holds for $x=\frac{y}{5}=\frac{z}{7}.$

Therefore, the maximum value of $\sqrt{x}+5\sqrt{y}+7\sqrt{z}$ is $75$ .

[[wiki-Cauchy