a.
\(\left\{{}\begin{matrix}a;b\ge0\\a+b< 1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}0\le a< 1\\0\le b< 1\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}a\left(1-a\right)\ge0\\b\left(1-b\right)\ge0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a\ge a^2\\b\ge b^2\end{matrix}\right.\)
\(\Rightarrow a^2+b^2\le a+b< 1\) (đpcm)
b.
- Với \(x\ge5\Rightarrow x+y\ge5+5=10\) hiển nhiên đúng
- Với \(y\ge6\Rightarrow x+y\ge4+6=10\) (đúng)
- Với \(\left\{{}\begin{matrix}4\le x< 5< 7\\5\le y\le6\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}\left(x-4\right)\left(7-x\right)\ge0\\\left(y-5\right)\left(6-y\right)\ge0\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}11x\ge x^2+28\\11y\ge y^2+30\end{matrix}\right.\)
\(\Rightarrow11\left(x+y\right)\ge x^2+y^2+58\ge52+58=110\)
\(\Rightarrow x+y\ge10\)
Dấu "=" xảy ra khi \(\left(x;y\right)=\left(4;6\right)\)
