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Có: \(\sqrt{2015}< \sqrt{2016}\)
=>\(\frac{1}{\sqrt{2015}}>\frac{1}{\sqrt{2016}}\)
=>\(\frac{1}{\sqrt{2015}}-\frac{1}{\sqrt{2016}}>0\)
=>\(\sqrt{2015}+\sqrt{2016}+\frac{1}{\sqrt{2015}}-\frac{1}{\sqrt{2016}}>\sqrt{2015}+\sqrt{2016}\)
=>\(\left(\sqrt{2015}+\frac{1}{\sqrt{2015}}\right)+\left(\sqrt{2016}-\frac{1}{\sqrt{2016}}\right)>\sqrt{2015}+\sqrt{2016}\)
=>\(\frac{2016}{\sqrt{2015}}+\frac{2015}{\sqrt{2016}}>\sqrt{2015}+\sqrt{2016}\)
Áp dụng bđt \(\frac{\sqrt{a}+\sqrt{b}}{2}< \sqrt{\frac{a+b}{2}}\) với a > 0; b > 0; a \(\ne\) b ta có:
\(\frac{\sqrt{2016}+\sqrt{2014}}{2}< \sqrt{\frac{2016+2014}{2}}\)
\(\Rightarrow\frac{\sqrt{2016}+\sqrt{2014}}{2}< \sqrt{\frac{4030}{2}}\)
\(\Rightarrow\frac{\sqrt{2016}+\sqrt{2014}}{2}< \sqrt{2015}\)
\(\Rightarrow\sqrt{2016}+\sqrt{2014}< 2.\sqrt{2015}\)
\(\Rightarrow\sqrt{2016}-\sqrt{2015}< \sqrt{2015}-\sqrt{2014}\)
Ta có : \(\frac{2014}{2015}+\frac{2015}{2016}+\frac{2016}{2014}=\left(\frac{2014}{2015}+\frac{1}{2014}\right)+\left(\frac{2015}{2016}+\frac{1}{2014}\right)+\frac{2014}{2014}\)
Mà : \(\left(\frac{2014}{2015}+\frac{1}{2014}\right)>1;\left(\frac{2015}{2016}+\frac{1}{2014}\right)>1;\frac{2014}{2014}=1\)
Nên : \(\frac{2014}{2015}+\frac{2015}{2016}+\frac{2016}{2014}=\left(\frac{2014}{2015}+\frac{1}{2014}\right)+\left(\frac{2015}{2016}+\frac{1}{2014}\right)+\frac{2014}{2014}\)\(>1+1+1=3\)
Ta có:\(\frac{2014}{2015}+\frac{2015}{2016}+\frac{2016}{2014}=\left(\frac{2014}{2015}+\frac{1}{2014}\right)\)\(+\left(\frac{2015}{2016}+\frac{1}{2014}\right)+\frac{2014}{2014}\)
Mà:\(\left(\frac{2014}{2015}+\frac{1}{2014}\right)>1:\left(\frac{2015}{2016}+\frac{1}{2014}\right)>\)\(1:\frac{2014}{2014}=1\)
Nên:\(\frac{2014}{2015}+\frac{2015}{2016}+\frac{2016}{2014}=\left(\frac{2014}{2015}+\frac{1}{2014}\right)\)\(+\left(\frac{2015}{2016}+\frac{1}{2014}\right)+\frac{2014}{2014}>1+1+1=3\)
\(\sqrt{2016}-\sqrt{2015}=\dfrac{1}{\sqrt{2016}+\sqrt{2015}}\)
\(\sqrt{2015}-\sqrt{2014}=\dfrac{1}{\sqrt{2015}+\sqrt{2014}}\)
mà \(\sqrt{2016}+\sqrt{2015}>\sqrt{2014}+\sqrt{2015}\)
nên \(\sqrt{2016}-\sqrt{2015}< \sqrt{2015}-\sqrt{2014}\)
Ta có :
\(x=\frac{2016^{2017}+1}{2016^{2016}+1}\)
\(\frac{1}{2016}x=\frac{2016^{2017}+1}{2016^{2017}+2016}=\frac{2016^{2017}+2016-2015}{2016^{2017}+2016}\)
\(\Rightarrow\frac{1}{2006}x=1-\frac{2015}{2016^{2017}+2016}\)
Ta lại có :
\(y=\frac{2016^{2016}+1}{2016^{2015}+1}\)
\(\Rightarrow\frac{1}{2016}y=\frac{2016^{2016}+1}{2016^{2016}+2016}=\frac{2016^{2016}+2016-2015}{2016^{2016}+2016}\)
\(\Rightarrow\frac{1}{2016}y=1-\frac{2015}{2016^{2016}+2016}\)
Mà \(\frac{2015}{2016^{2017}+2016}< \frac{2015}{2016^{2016}+2016}\)(so sánh mẫu)
\(\Rightarrow1-\frac{2015}{2016^{2017}+2016}>1-\frac{2015}{2016^{2016}+2016}\)
\(\Rightarrow\frac{1}{2016}x>\frac{1}{2016}y\)
\(\Rightarrow x>y\)
DÀI QUÁ KHÔNG TÍNH ĐƯỢC. CÁI NÀY CÓ MÀ ĐI HỎI THẦN ĐỒNG VỀ MÔN TOÁN ĐI
Ta có :2016>2015 nên căn 2016>căn 2015