\(x^3+y^3=8-6xy\)

\(\Leftrightarrow\left(x+y\right)^3-3xy\left(x+y\right)-8+6xy=0\)

\(\Leftrightarrow\left(x+y\right)^3-2^3-3xy\left(x+y-2\right)=0\)

\(\Leftrightarrow\left(x+y-2\right)\left[\left(x+y\right)^2+2\left(x+y\right)+4\right]-3xy\left(x+y-2\right)=0\)

\(\Leftrightarrow\left(x+y-2\right)\left(x^2+y^2-xy+2x+2y+4\right)=0\)

\(\Leftrightarrow\left(x+y-2\right)\left(2x^2+2y^2-2xy+4x+4y+8\right)=0\)

\(\Leftrightarrow\left(x+y-2\right)\left[\left(x-y\right)^2+\left(x+2\right)^2+\left(y+2\right)^2\right]=0\)

\(\Rightarrow\left[{}\begin{matrix}x+y-2=0\\\left(x-y\right)^2=\left(x+2\right)^2=\left(y+2\right)^2=0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x+y=2\\x=y=-2\left(loại\right)\end{matrix}\right.\)

\(x+y=1\Rightarrow y=1-x\)

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

\(P=2x^2-2x+1=\dfrac{1}{2}\left(2x-1\right)^2+\dfrac{1}{2}\ge\dfrac{1}{2}\)

\(P_{min}=\dfrac{1}{2}\) khi \(x=y=\dfrac{1}{2}\)

Từ giả thiết ta có:

\(x+y=3\left(\sqrt{x+1}+\sqrt{y+2}\right)\le3\sqrt{2\left(x+y+3\right)}\)

\(\Leftrightarrow P\le3\sqrt{2\left(P+3\right)}\)

\(\Leftrightarrow\left\{{}\begin{matrix}P\ge0\\18P+54\ge P^2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}P\ge0\\P^2-18P-54\le0\end{matrix}\right.\)

\(\Leftrightarrow0\le P\le9+3\sqrt{15}\)

\(\Rightarrow maxP=9+3\sqrt{15}\Leftrightarrow\left(x;y\right)=\left(\dfrac{10+3\sqrt{15}}{2};\dfrac{8+3\sqrt{15}}{2}\right)\)

 Ta có :

\(x^2+y^2+xy=3\)

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

\(\Rightarrow \left(x+y\right)^2=3+xy\)

hay \(S^2=3+xy\le3+\frac{\left(x+y\right)^2}{4}=3+\frac{S^2}{4}\)

\(\Rightarrow S^2\le3+\frac{S^2}{4}\)

\(\Rightarrow S^2\le4\)

\(\Rightarrow-2\le S\le2\)

GTLN của S = 2

Đáp án D

Đáp án D.

Thì ra cx có ng k hiểu thầy nói gì giống mình

x^2+y^2=(x+y)^2-2xy

=5^2-2*3

=25-6

=19

x^3+y^3=(x+y)^3-3xy(x+y)

=5^3-3*3*5

=125-9*5

=80

(x-y)^2=(x+y)^2-4xy=5^2-4*3=13

=>\(x-y=\sqrt{13}\)