TITLE: How do you prove inequalities like $x^2 + xy + y^2 \ge 0$?
QUESTION [3 upvotes]: This is what I am asking. I do not want you to prove these inequalities for me; instead, I would like you to teach me a general method with which I can prove the inequality:
$$x^2 + xy + y^2 \ge 0$$
If you could prove a similar inequality to illustrate the general method, I will be thankful.

REPLY [0 votes]: Since

$$\begin{equation}\begin{split} y^{2}+ yx+ x^{2}= (- y+ x)^{2}+ 3yx= (y+ x)^{2}- yx \end{split}\end{equation}$$

Since

$$\begin{equation}\begin{split} y^{2}+ yx+ x^{2}= \frac{y^{3}- x^{3}}{y- x}\geqq 0\because \lim_{y\rightarrow x+ 0^{+}}\frac{y^{3}- x^{3}}{y- x}= 3x^{2}\geqq 0, \lim_{y\rightarrow x+ 0^{-}}\frac{y^{3}- x^{3}}{y- x}= 3x^{2}\geqq 0 \end{split}\end{equation}$$
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