Question

Tìm GTLN của f_(x,y) =\(\dfrac{xy^2+y^2\left(y^2-x\right)+1}{x^2y^4+2y^4+x^2+2}\)

Answer 1

\(f\left(x,y\right)=\dfrac{xy^2+y^4-xy^2+1}{y^4\left(x^2+2\right)+x^2+2}=\dfrac{y^4+1}{\left(x^2+2\right)\left(y^4+1\right)}=\dfrac{1}{x^2+2}\)

Do \(x^2+2\ge2\) \(\Rightarrow\dfrac{1}{x^2+2}\le\dfrac{1}{2}\)

\(\Rightarrow f\left(x,y\right)_{max}=\dfrac{1}{2}\) khi \(x^2+2=2\Leftrightarrow x=0\)