Question

Tìm GTNN của biểu thức sau:

E=5x²+y²+10+4xy-14x-6y

Answer 1

\(E=5x^2+y^2+10+4xy-14x-6y\)

\(=\left(4x^2+y^2+4xy\right)-12x-6y+9+x^2-2y+1\)

\(=\left(2x+y\right)^2-6\left(2x+y\right)+9+\left(x-1\right)^2\)

\(=\left(2x+y-3\right)^2+\left(x-1\right)^2\ge0\)

\(\Rightarrow E_{Min}=0\)

\("="\Leftrightarrow x=y=1\)

Answer 2

Ta có E= \(\left(4x^2+y^2+9-6y-12x+4xy\right)+\left(x^2-2x+1\right)\)

=\(\left(2x+y-3\right)^2+\left(x-1\right)^2\)

Vì \(\left(2x+y-3\right)^2+\left(x-1\right)^2\) >= 0

=>E>=0 =>GTNN của E=0 khi: \(x-1=0\) =>\(x=1\)

\(2x+y-3=0\) =>\(2x+y=3\)

=> \(2+y=3\) => \(y=1\)