Question

Tìm giá trị lớn nhất hoặc giá trị nhỏ nhất: \(D=x^4-8xy-x^3y+x^2y^2-xy^3+y^4+2000\)

Answer 1

\(D=x^3\left(x-y\right)-y^3\left(x-y\right)+\left(x^2y^2-8xy+16\right)+1984\)

\(D=\left(x-y\right)\left(x^3-y^3\right)+\left(xy-4\right)^2+1984\)

\(D=\left(x-y\right)^2\left(x^2+xy+y^2\right)+\left(xy-4\right)^2+1984\)

\(D=\left(x-y\right)^2\left[\left(x+\frac{y}{2}\right)^2+\frac{3y^2}{4}\right]+\left(xy-4\right)^2+1984\ge1984\)

\(D_{min}=1984\) khi \(x=y=\pm2\)