\(\sqrt{a}+\sqrt{b}=1\)
\(\Rightarrow\left(\sqrt{a}+\sqrt{b}\right)^2=1\)
\(\Rightarrow a+b+2\sqrt{a}.\sqrt{b}=1\)
\(\Rightarrow a+b+2\sqrt{ab}=1\)
Mà: \(\left(a+b\right)+2\sqrt{ab}\ge2\sqrt{\left(a+b\right).2\sqrt{ab}}\)
\(\Rightarrow1\ge2\sqrt{\left(a+b\right).2\sqrt{ab}}\)
\(\Rightarrow\frac{1}{2}\ge\sqrt{\left(a+b\right).2\sqrt{ab}}\)
\(\Rightarrow\frac{1}{4}\ge\left(a+b\right).2\sqrt{ab}\)
\(\Rightarrow\frac{1}{8}\ge\left(a+b\right)\sqrt{ab}\)
\(\Rightarrow\frac{1}{64}\ge\left[\left(a+b\right).\sqrt{ab}\right]^2\)
\(\Rightarrow\frac{1}{64}\ge\left(a+b\right)^2.ab\)
\(\Rightarrow P=\left(a+b\right)^2.ab\le\frac{1}{64}\)