<s>[INST] Finding maxima and minima of $f(x,y)= x^4 + y^4 - 2x^2 - 2y^2 + 4xy$ For $f(x,y)=x^4+y^4-2x^2-2y^2+4xy$, I need to find maxima or minima. There are three critical points: $(0,0),(\sqrt2, -\sqrt2),(-\sqrt2,\sqrt2)$
So at $(\sqrt2, -\sqrt2)$, $f$ has minimum value, $-8$ and at $(-\sqrt2,\sqrt2),$ it has same minimum value, $-8$. At $(0,0)$, after inspecting, I get that it has neither maxima or minima. But when we substitute $(0,0),$ we get $0\le f \le8,$ so should not we get $(0,0)$ as a point of maxima?
 [/INST] It's obvious that a maximal value does not exist.
We'll prove that $-8$ is a minimal value.
Let $x=\sqrt2a$ and $y=-\sqrt2b$.
Hence, we need to prove that
$$4a^4+4b^4-4a^2-4b^2-8ab+8\geq0$$ or
$$a^4+b^4+2\geq(a+b)^2,$$
which is AM-GM and C-S:
$$a^4+b^4+2\geq2\sqrt{(a^4+1)(1+b^4)}\geq2(a^2+b^2)\geq(a+b)^2$$
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