<s>[INST] Find minimum value $E(x, y) = x^2 + y^2 -6x -10y$, where $x^2 + y^2 - 2y \le 0$ I am given the expression:
$$E(x, y) = x^2 + y^2 -6x -10y$$
And I have to find the minimum value of $E(x, y)$ for $(x, y) \in D$ where:
$$D = \{ (x,y) \in \mathbb{R}^2 \hspace{0.25cm} | \hspace{0.25cm} x^2 + y^2 -2y \le 0 \}$$
Doing the following:
$$\hspace{5.8cm} x^2+y^2-2y \le 0 \hspace{5cm} |+1$$
$$x^2+(y-1)^2 \le 1$$
So I have to find the minimum value of $E(x,y)$ where $(x, y)$ is from the circle $x^2 + (y-1)^2 \le 1$.
I don't see how I should approach this.
 [/INST] For $(x,y)=\left(\frac{3}{5},\frac{9}{5}\right)$ we'll get a value $-18$.
We'll prove that it's a minimal value.
Indeed, it's enough to prove that:
$$x^2+y^2-6x-10y\geq-18+4(2y-x^2-y^2)$$ or
$$5x^2-6x+5y^2-18y+18\geq0$$ or
$$5\left(x-\frac{3}{5}\right)^2+5\left(y-\frac{9}{5}\right)^2\geq0$$ and we are done!
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