Cristian Calude, proba pe echipe, R.IV, P.III

[Laurian Filip]

Fie expresia:
\( E(x,y)=x^2+6y^2-2xy+6x-16y+18 \)
Sa se determine \( \min_{x,y \in \mathbb{R}} E(x,y) \)

[alex2008]

\( E = x^2 + 6y^2 - 2xy + 6x - 16y + 18 = (x - y)^2 + 6(x - y) + 9 + 5(y^2 - 2y + 1) + 4 \)

\( = (x - y + 3)^2 + 5(y - 1)^2 + 4 \ge 4 \)

\( \Rightarrow E_{min} = 4 \Longleftrightarrow x = - 2;y = 1 \)