Ta có:
\(M=\dfrac{\left(a+b\right)^2-\left(a^2+b^2\right)}{2}+\dfrac{1}{a+b}=\dfrac{\left(a+b\right)^2-1}{2}+\dfrac{1}{a+b}\)
\(M=\dfrac{\left(a+b\right)^2}{2}+\dfrac{1}{2\left(a+b\right)}+\dfrac{1}{2\left(a+b\right)}-\dfrac{1}{2}\)
\(M\ge3\sqrt[3]{\dfrac{\left(a+b\right)^2}{8\left(a+b\right)^2}}-\dfrac{1}{2}=1\)
\(M_{min}=1\) khi \(a+b=1\Leftrightarrow\left(a;b\right)=\left(0;1\right);\left(1;0\right)\)
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