Find a minimum and a maximum of expression $E(x,y,z)=x-2y+z$

Question

Find minimum and maximum of $E(x,y,z)=x-2y+z$ where $2x^2+y^2+z^2=3$.

For finding the maximum of the expression I first thought using the inequality of Cauchy-Buniakowski-Schwarz: $$\boxed{(ax+by+cz)^2\leq(a^2+b^2+c^2)(x^2+y^2+z^2)}$$

What I get is:

$$(E(x,y,z))^2\leq(1^2+(-2)^2+1^2)(x^2+y^2+z^2)\implies E(x,y,z)\leq \sqrt{6(3-x^2)}$$

But I don't know how to use it. Maybe should I use the inequality in a different manner?

Answer 1

$$E(x,y,z)\leq \Big{|}{1\over \sqrt{2}}\cdot \sqrt{2}x+(-2)\cdot y+1\cdot z\Big{|}\leq \sqrt{({1\over 2}+4+1)(2x^2+y^2+z^2)}$$

So $$ |E| \leq {\sqrt{33\over 2}}$$ so that $$ -{\sqrt{33\over 2}}\leq E\leq {\sqrt{33\over 2}}$$

If $\max $ is acchieved at $(x_0,y_0,z_0)$ then minimum is at $(-x_0,-y_0,-z_0)$.