<s>[INST] The smallest value of the expression $4x^2y^2+x^2+y^2-2xy+x+y+1$ What is the smallest value that $4x^2y^2+x^2+y^2-2xy+x+y+1$ can take with real numbers $x$ and $y$?
I suspect the following transformation can be done: $(2xy-1/2)^2 + (x+1/2)^2 + (y+1/2)^2 + 1/4$.
 [/INST] After $(2xy-1/2)^2 + (x+1/2)^2 + (y+1/2)^2 + 1/4$, use a change of variables $x - 1/2 = u, y - 1/2 = v$, and $(2xy-1/2) = 2(u + 1/2)(v + 1/2) - 1/2$ $ = 2(uv + u/2 + v/2 + 1/4) - 1/2$ to get:
$$(2uv+u+v)^2 + u^2 + v^2 + 1/4$$
$$=(2uv+u+v)^2 + (u+v)^2 - 2uv + 1/4$$
$$= (p+q)^2 + q^2 - p + 1/4$$
where $p=2uv, q = u+v$.
For $u, v$ to be real numbers, $q^2 - p ≥ 0$ as $u^2+v^2 ≥ 0$ for all real numbers $u,v$. Thus  $(p+q)^2 + q^2 - p + 1/4 ≥ (p + q)^2 + 1/4 ≥ 1/4$.
This minimum is attained in the original expression when $x = y = -1/2 $.
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