Câu a : Ta có :
\(\sqrt{2016}-\sqrt{2015}=\dfrac{1}{\sqrt{2016}+\sqrt{2015}}\)
\(\sqrt{2017}-\sqrt{2016}=\dfrac{1}{\sqrt{2017}+\sqrt{2016}}\)
Mà : \(\sqrt{2016}+\sqrt{2015}< \sqrt{2017}+\sqrt{2016}\) \(\Rightarrow\dfrac{1}{\sqrt{2016}+\sqrt{2015}}>\dfrac{1}{\sqrt{2017}+\sqrt{2016}}\)
Vậy : \(\sqrt{2016}-\sqrt{2015}>\sqrt{2017}-\sqrt{2016}\)
Câu b : Ta có : \(P=\left(\sqrt{a}+\sqrt{b}\right)\Rightarrow P^2=\left(\sqrt{a}+\sqrt{b}\right)^2\)
Theo BĐT Bu - nhi - a - cốp xki ta có :
\(P^2=\left(\sqrt{a}+\sqrt{b}\right)^2\le\left(1^2+1^2\right)\left(a+b\right)=2.1008=2016\)
\(\Rightarrow P\le\sqrt{2016}\)
Vậy GTLN của P là \(\sqrt{2016}\) khi \(a=b=504\)
