algebra-precalculuscauchy-sequences
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.
This question is an old problem posed on Brilliant.
You can view it here, along with the solution by the problem creator Zi Song.
Let $\vec a=(2,-5), \vec b=(x,y)$.
$2x-5y=\vec a\cdot \vec b=\sqrt{2^2+(-5)^2}\cdot\sqrt{x^2+y^2}\cos\theta\le\sqrt{29}$ where $\theta$ is the angle between the two vectors.
Best Answer
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 $(x,y,z)=\frac{1}{17}\left(9,12,8\right).$