Áp dụng bđt bunhicopxki ta có:
\(\left(\sqrt{a\left(4a+5b\right)}+\sqrt{b\left(4b+5a\right)}\right)^2\le\left(a+b\right)\left(4a+5b+4b+5a\right)=9\left(a+b\right)^2\)
=> \(\sqrt{a\left(4a+5b\right)}+\sqrt{b\left(4b+5a\right)}\le3\left(a+b\right)\)
=>\(P\ge\frac{a+b}{3\left(a+b\right)}=\frac{1}{3}\)
dấu "=" xảy ra khi a=b