<s>[INST] Find the maximum and minimum value of $2x^2 + y^2 + z^2$ subject to $x + y + z = 10$. I am working on inequality problems for mathematical olympiads. I have come across this problem:

Find the maximum and minimum value of $2x^2 + y^2 + z^2$ subject to $x + y + z = 10$, with $x,y,z \ge 0 $

I found the minimum using Cauchy-Schwarz: $$100 = \left(\frac 1 {\sqrt 2} \left(\sqrt 2 x\right) + y + z\right)^2 \le \left(\left(1/\sqrt 2\right)^2 + 1^2 + 1^2\right) \left(\left(\sqrt 2 x\right)^2 + y^2 + z^2\right)=2.5\left(2x^2 + y^2 + z^2\right)$$
so $2x^2 + y^2 + z^2 \ge 40$ with equality if $x=2$ and $y=z=4$.
I am not sure how to find the maximum though. Could I have a hint? Ideally I would like to do it without using calculus, though if it is easy to do with calculus feel free to share the method.
 [/INST] Substitute the value of $z$ in the expression: $$2x^2 +y^2 +(x+y-10)^2 $$ As increasing $x$ and $y$ increases the value of this expression, the maximum occurs when $x+y$ is maximum, i.e. $x+y=10$: $$2x^2 +(10-x)^2 = 3x^2 -20x +100$$
This is a parabola which attains its maximum value on $[0,10]$ at $x=10$, which is $200$.
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