Ta có: \(\left(\sqrt{2018}+\sqrt{2020}\right)^2=2018+2020+2\sqrt{2018.2020}\)
\(=4038+2\sqrt{\left(2019-1\right)\left(2019+1\right)}< 4038+2\sqrt{2019^2}\)
\(=4038+4038=8076\) (1)
Ta cũng có: \(\left(2\sqrt{2019}\right)^2=4.2019=8076\) (2)
Từ (1) và (2) \(\Rightarrow\sqrt{2018}+\sqrt{2020}< 2\sqrt{2019}\)
Xét \(\left(\sqrt{2018}+\sqrt{2020}\right)^2=2018+2020+2\sqrt{2018.2020}\)
\(=2019+2019+2\sqrt{\left(2019+1\right)\left(2019-1\right)}\)
\(=2.2019+2\sqrt{2019^2-1}\)
Có \(\sqrt{2019^2-1}< \sqrt{2019^2}\Rightarrow2\sqrt{2019^2-1}< 2.2019\)
\(\Rightarrow2.2019+2\sqrt{2019^2-1}< 2.2019+2.2019=4.2019=\left(2\sqrt{2019}\right)^2\)
\(\Rightarrow\left(\sqrt{2018}+\sqrt{2020}\right)^2< \left(2\sqrt{2019}\right)^2\)
\(\Leftrightarrow\sqrt{2018}+\sqrt{2020}< 2\sqrt{2019}\)
