Let $x$, $y$, and $z$ be real numbers such that $x^2 + y^2 + z^2 = 1.$ Find the maximum value of $9x+12y+8z.$ Let $x$, $y$, and $z$ be real numbers such that $x^2 + y^2 + z^2 = 1.$ Find the maximum value of $9x+12y+8z.$

We aren't given anything other than what $x^2+y^2+z^2$ is.  How can I manipulate that equation into something useful?  I'm stuck.  Any solutions are highly appreciated.

By the Cauchy-Schwarz inequality
$$9x+12y+8z \color{red}\leq \sqrt{9^2+12^2+8^2}\sqrt{x^2+y^2+z^2} = \color{red}{17} $$
and equality is achieved at $(x,y,z)=\frac{1}{17}\left(9,12,8\right).$
