\(a+b\le\sqrt{2\left(a^2+b^2\right)}=\sqrt{2}\)
Dấu "=" xảy ra khi \(a=b=\frac{1}{\sqrt{2}}\)
Mặt khác do \(a^2+b^2=1\Rightarrow\left\{{}\begin{matrix}a^2\le1\\b^2\le1\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}0\le a\le1\\0\le b\le1\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}a^2\le a\\b^2\le b\end{matrix}\right.\) \(\Rightarrow a+b\ge a^2+b^2=1\)
Dấu "=" xảy ra khi \(\left(a;b\right)=\left(1;0\right);\left(0;1\right)\)