Question

Tìm GTNN, GTLN của

P = x² + y² = xy + 16

Answer 1

\(P=x^2+y^2=xy+16\) \(\Rightarrow2P=2\left(x^2+y^2\right)=2xy+32\)

\(\Rightarrow3\left(x^2+y^2\right)=\left(x+y\right)^2+32\ge32\)

\(\Rightarrow x^2+y^2\ge\frac{32}{3}\Rightarrow P\ge\frac{32}{3}\)

\(P=xy+16\le\frac{x^2+y^2}{2}+16\)

\(\Rightarrow x^2+y^2\le\frac{x^2+y^2}{2}+16\)

\(\Rightarrow\frac{x^2+y^2}{2}\le16\Rightarrow x^2+y^2\le32\)

\(\Rightarrow P\le32\)