Chắc đề ghi nhầm ngoặc sau (2 mẫu kia thực chất giống nhau, lẽ ra phải là \(\dfrac{1}{\sqrt{a+3b}}+\dfrac{1}{\sqrt{3a+b}}\)
\(VT=\sqrt{\dfrac{a}{a+3b}}+\sqrt{\dfrac{a}{3a+b}}+\sqrt{\dfrac{b}{a+3b}}+\sqrt{\dfrac{b}{3a+b}}\)
\(=\sqrt{\dfrac{a}{a+b}.\dfrac{a+b}{a+3b}}+\sqrt{\dfrac{1}{2}.\dfrac{2a}{3a+b}}+\sqrt{\dfrac{1}{2}.\dfrac{2b}{a+3b}}+\sqrt{\dfrac{b}{a+b}.\dfrac{a+b}{3a+b}}\)
\(\le\dfrac{1}{2}\left(\dfrac{a}{a+b}+\dfrac{a+b}{a+3b}\right)+\dfrac{1}{2}\left(\dfrac{1}{2}+\dfrac{2a}{3a+b}\right)+\dfrac{1}{2}\left(\dfrac{1}{2}+\dfrac{2b}{a+3b}\right)+\dfrac{1}{2}\left(\dfrac{b}{a+b}+\dfrac{a+b}{3a+b}\right)\)
\(=\dfrac{1}{2}\left(1+\dfrac{a+b}{a+b}+\dfrac{a+3b}{a+3b}+\dfrac{3a+b}{3a+b}\right)=2\)
Dấu "=" xảy ra khi \(a=b\)
