Tìm GTNN: A = x^2 − 2xy + 2y^2 + 2x − 10y +17

Câu 22: Tìm GTNN: A = x² − 2xy + 2y² + 2x − 10y +17.

Trả lời

A = x² − 2xy + 2y² + 2x − 10y +17

= (x² − 2xy + y²) + 2(x − y) + 1 + (y² − 8y +16)

= (x − y)² + 2(x − y) + 1 + (y − 4)²

= (x − y + 1)² + (y − 4)² ≥ 0.

Vậy GTNN của A bằng 0 khi $\left\{\begin{array}{l}x-y+1=0\\ y-4=0\end{array}\right.\iff \left\{\begin{array}{l}x=3\\ y=4\end{array}\right.$ .