A=x³+y³=(x+y)(x²-xy+y²)

=(x+y)²\(\ge\)0

Dấu "=" xảy ra khi x=-y

\(\hept{\begin{cases}x+y\le2\\x^2+y^2+xy=3\end{cases}\Leftrightarrow\hept{\begin{cases}x+y=2-a\\x^2+y^2+xy=3\end{cases}\left(a\ge0\right)}}\)

Do đó: \(\hept{\begin{cases}x+y=2-a\\xy=\left(2-a\right)^2-3\end{cases}}\)

Điều kiện có nghiệm là: \(\Delta=S^2-4P\ge0\)và a>=0 nên 0 =<a =< 4

Ta có: \(T=x^2+y^2+xy-2xy=9-2\left(2-a\right)^2\)

=> \(Min_T=1\)khi x=1 và y=1 hoặc x=-1; y=-1

\(Max_T=9\)khi \(x=\sqrt{3};y=-\sqrt{3}\)hoặc \(x=-\sqrt{3};y=\sqrt{3}\)

pro rồi thì bạn cần gì mình giải nhỉ

??

\(A=x-2y+3\Rightarrow x=A+2y-3\)

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

\(\Leftrightarrow8y^2+\left(5A-15\right)y+A^2-6A+8=0\)

\(\Delta=\left(5A-15\right)^2-32\left(A^2-6A+8\right)\ge0\)

\(\Leftrightarrow-7A^2+42A-31\ge0\)

\(\Rightarrow\dfrac{21-4\sqrt{14}}{7}\le A\le\dfrac{21+4\sqrt{14}}{7}\)

\(A=\dfrac{x^3+y^3+4}{xy+1}\ge\dfrac{x^3+y^3+4}{\dfrac{x^2+y^2}{2}+1}=\dfrac{x^3+y^3+4}{2}=\dfrac{\dfrac{1}{2}\left(x^3+x^3+1\right)+\dfrac{1}{2}\left(y^3+y^3+1\right)+3}{2}\)

\(\ge\dfrac{\dfrac{3}{2}\left(x^2+y^2\right)+3}{2}=3\)

\(A_{min}=3\) khi \(x=y=1\)

Do \(x^2+y^2=2\Rightarrow\left\{{}\begin{matrix}x\le\sqrt{2}\\y\le\sqrt{2}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x^3\le\sqrt{2}x^2\\y^3\le\sqrt{2}y^2\end{matrix}\right.\)

\(\Rightarrow A\le\dfrac{\sqrt{2}\left(x^2+y^2\right)+4}{xy+1}=\dfrac{4+2\sqrt{2}}{xy+1}\le\dfrac{4+2\sqrt{2}}{1}=4+2\sqrt{2}\)

\(A_{max}=4+2\sqrt{2}\) khi \(\left(x;y\right)=\left(0;\sqrt{2}\right);\left(\sqrt{2};0\right)\)