Question

1. Tìm GTNN của biểu thức:

A= x² + 2y²-2xy-4x-6y-3

B= 4x²+2y²-4xy+4x+6y+1

Answer 1

*\(A=x^2+2y^2-2xy-4x-6y-3\)

\(A=x^2-2x\left(y+2\right)+\left(y^2+4y+4\right)+\left(y^2-10y+25\right)-32\)

\(A=x^2-2x\left(y+2\right)+\left(y+2\right)^2+\left(y-5\right)^2-32\)

\(A=\left(x-y-2\right)^2+\left(y-5\right)^2-32\ge-32\)

\(\Rightarrow Min_A=-32\Leftrightarrow x=7;y=5\)

* \(B=4x^2+2y^2-4xy+4x+6y+1\)

\(B=\left(2x\right)^2-\left(4xy+4x\right)+\left(y^2-2y+1\right)+\left(y^2+8y+16\right)-16\)\(B=\left(2x\right)^2-2.2x\left(y-1\right)+\left(y-1\right)^2+\left(y+4\right)^2-16\)\(B=\left(2x-y+1\right)^2+\left(y+4\right)^2-16\ge-16\)

\(\Rightarrow Min_B=-16\Leftrightarrow x=-\dfrac{5}{2};y=-4\)