\(\left\{{}\begin{matrix}\sqrt{a}=x\ge0\\\sqrt{b}=y\ge0\end{matrix}\right.\) \(\Rightarrow x+y=1\)
\(P=x^3+y^3=\left(x+y\right)\left(x^2+y^2-xy\right)=x^2+y^2-xy\)
\(P=\left(x+y\right)^2-3xy=1-3xy\)
Do \(\left\{{}\begin{matrix}x\ge0\\y\ge0\end{matrix}\right.\) \(\Rightarrow xy\ge0\Rightarrow P\le1\Rightarrow P_{max}=1\) khi \(\left(x;y\right)=\left(1;0\right);\left(0;1\right)\) hay \(\left(a;b\right)=\left(1;0\right);\left(0;1\right)\)
Lại có \(xy\le\frac{\left(x+y\right)^2}{4}=\frac{1}{4}\Rightarrow P\ge1-3.\frac{1}{4}=\frac{1}{4}\)
\(\Rightarrow P_{min}=\frac{1}{4}\) khi \(x=y=\frac{1}{2}\) hay \(a=b=\frac{1}{4}\)
\(\Rightarrow m^2+n^2=1+\frac{1}{16}=\frac{17}{16}\)
