<s>[INST] Inequality: $x^2+y^2+z^2 \geq \sqrt{2}x(z+y)$ How can I prove the following inequality:
$$x^2+y^2+z^2 \geq \sqrt{2}x(z+y)?$$
Thanks!
 [/INST] Obviously, is enough to prove for $x\ge0$, $y\ge0$, $z\ge0$. By homogeneity, can be supposed wlog that $x^2+y^2+z^2=1$ Using Lagrange multipliers with the problem
$$f(x,y,z)=\sqrt{2}x(z+y),\qquad x^2+y^2+z^2=1,$$
we have:
$$\sqrt{2}(z+y)=2\lambda x,$$
$$\sqrt{2}x=2\lambda y,$$
$$\sqrt{2}x=2\lambda z,$$
with nontrivial solutions for $\lambda=0,\pm1$...
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