\(B=-5\left(3x+2\right)^4-\left(x+2y\right)^2+111\)
Ta có :
\(\left(3x+2\right)^4\ge0\Rightarrow-5\left(3x+2\right)^4\le0\left(1\right)\)
\(\left(x+2y\right)^2\ge0\Rightarrow-\left(x+2y\right)^2\le0\left(2\right)\)
Từ (1)(2) \(\Rightarrow-5\left(3x+2\right)^4-\left(x+2y\right)^2\le0\)
\(\Rightarrow-5\left(3x+2\right)^4-\left(x+2y\right)^2+111\le111\)
Dấu = xảy ra khi \(\left\{{}\begin{matrix}5\left(3x+2\right)^4=0\\\left(x+2y\right)^2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3x+2=0\\x+2y=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=-\dfrac{2}{3}\\-\dfrac{2}{3}+2y=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-\dfrac{2}{3}\\y=\dfrac{1}{3}\end{matrix}\right.\)
Vậy B đạt GTLN bằng 11 khi \(x=-\dfrac{2}{3};y=\dfrac{1}{3}\)
\(A=-5\left(3x+2\right)^4-\left(x+2y\right)^2+111\le111\)
Dấu "=" xảy ra khi:
\(\left\{{}\begin{matrix}-5\left(3x+2\right)^4=0\\-\left(x+2y\right)=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-\dfrac{2}{3}\\y=\dfrac{1}{3}\end{matrix}\right.\)