Question

Chứng minh rằng:

a)  x² + 4y² + z² - 2x - 6z + 8y + 15 > 0 với mọi x, y, z

b)  x² + 5y² + 2x - 4xy - 10y + 14 > 0 với mọi x, y

Answer 1

       \(x^2+4y^2+z^2-2x-6z+8y+15\)

\(=\left(x^2-2x+1\right)+\left(4y^2+8y+4\right)+\left(z^2-6z+9\right)+1\)

\(=\left(x-1\right)^2+4\left(y+1\right)^2+\left(z-3\right)^2+1>0\forall x;y\)

       \(x^2+5y^2+2x-4xy-10y+14\)

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

\(=\left(x-2y\right)^2+2\left(x-2y\right)+1+\left(y-3\right)^2+4\)

\(=\left(x-2y+1\right)^2+\left(y-3\right)^2+4>0\forall x;y\)

Chúc bạn học tốt.