ĐKXĐ: \(-3\le x\le3\)
\(\Leftrightarrow\left(y+1\right)^3+2\left(y+1\right)=\left(11-x^2\right)\sqrt{9-x^2}\)
Đặt \(\left\{{}\begin{matrix}\sqrt{9-x^2}=a\\y+1=b\end{matrix}\right.\)
\(\Rightarrow b^3+2b=\left(2+a^2\right)a=a^3+2a\)
\(\Leftrightarrow a^3-b^3+2\left(a-b\right)=0\)
\(\Leftrightarrow\left(a-b\right)\left(a^2+ab+b^2+2\right)=0\)
\(\Leftrightarrow a=b\Leftrightarrow\sqrt{9-x^2}=y+1\Rightarrow y=\sqrt{9-x^2}-1\)
\(\Rightarrow T=x-\left(\sqrt{9-x^2}-1\right)+2018=x-\sqrt{9-x^2}+2019\)
Đặt \(A=x-\sqrt{9-x^2}\)
Dễ thấy với \(x>0\Rightarrow A>0\); \(x< 0\Rightarrow A< 0\)
Do đó GTLN xảy ra khi \(x>0\); GTNN xảy ra khi \(x< 0\)
- Với \(x>0\Rightarrow A^2=9-2x\sqrt{9-x^2}\le9\Rightarrow A\le3\)
\(\Rightarrow T_{max}=3+2019=2022\) khi \(x=3\)
- Với \(x< 0\Rightarrow A^2=9-2x\sqrt{9-x^2}=9+2.\left(-x\right)\sqrt{9-x^2}\le9+\left(\left(-x\right)^2+9-x^2\right)=18\)
\(\Rightarrow A\ge-\sqrt{18}=-3\sqrt{2}\)
\(\Rightarrow T_{min}=-3\sqrt{2}+2019\) khi \(-x=\sqrt{9-x^2}\Leftrightarrow x=\frac{-3\sqrt{2}}{2}\)
