<s>[INST] Which point of the sphere $x^2+y^2+z^2=19$ maximize $2x+3y+5z$? Which point of the sphere $x^2+y^2+z^2=19$ maximize $2x+3y+5z$?

So I assume that there is a point maximizing $2x+3y+5z$. How can I calculate the exact value of this point?
 [/INST] By the Cauchy-Schwarz inequality
$$ 2x+3y+5z \leq \sqrt{x^2+y^2+z^2} \sqrt{2^2+3^2+5^2} = 19 \sqrt{38} $$
and equality is achieved if $(x,y,z)=\lambda(2,3,5)$, i.e. for $\lambda=\frac{1}{\sqrt{2}}$.
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