a) Áp dụng bất đẳng thức Cô-si :
\(a^2+x^2\ge2\left|ax\right|\)
\(b^2+y^2\ge2\left|by\right|\)
Cộng theo vế :
\(a^2+b^2+x^2+y^2\ge2\left(\left|ax\right|+\left|by\right|\right)\ge2\left|ax+by\right|\)
Dấu "=" xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}a=x\\b=y\\ax\ge0;by\ge0\end{matrix}\right.\)
b) \(\left|x\right|+\left|y\right|+\left|x-3\right|+\left|y-5\right|\)
\(=\left|x\right|+\left|3-x\right|+\left|y\right|+\left|5-y\right|\)
\(\ge\left|x+3-x\right|+\left|y+5-y\right|=\left|3\right|+\left|5\right|=8\)
Dấu "=" xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}x\left(3-x\right)\ge0\\y\left(5-y\right)\ge0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}0\le x\le3\\0\le y\le5\end{matrix}\right.\)
với x, y, z > 0.
