\(Q=\frac{2010,2011}{2014,2015}+\frac{2012,2013}{2014,2015}.\frac{1}{x^2}-\frac{2}{2014,2015}.\frac{1}{x}\)

\(=\frac{2010,2011}{2014,2015}+\frac{2012,2013}{2014,2015}\left(\frac{1}{x^2}-2.\frac{1}{x}.\frac{1}{2012,2013}+\frac{1}{\left(2012,2013\right)^2}\right)-\frac{1}{2012,2013}.\frac{1}{2014,2015}\)

\(=\frac{2010,2011}{2014,2015}-\frac{1}{2012,2013}.\frac{1}{2014,2015}+\frac{2012,2013}{2014,2015}\left(\frac{1}{x}-\frac{1}{2012,2013}\right)^2\)

\(\ge\frac{2010,2011}{2014,2015}-\frac{1}{2012,2013}.\frac{1}{2014,2015}\)

Dấu "=" xảy ra khi và chỉ khi \(x=2012,2013\)

Vậy GTNN của biểu thức là .....

ĐKXĐ:

\(x\ge0\text{ và }\sqrt{x}-1\ge0\text{ và }x+1\ne0\)

\(\Leftrightarrow x\ge1\text{ và }x\ne-1\)

\(\left(\frac{x\sqrt{x}}{x\sqrt{x}+x+\sqrt{x}+1}+\frac{1}{x+1}\right):\frac{\sqrt{x}-1}{x+1}\)

\(=\left(\frac{x\sqrt{x}}{\left(\sqrt{x}+1\right)\left(x+1\right)}+\frac{1}{x+1}\right).\frac{x+1}{\sqrt{x}+1}\)

\(=\left(\frac{x\sqrt{x}}{\left(\sqrt{x}+1\right)\left(x+1\right)}+\frac{\sqrt{x}+1}{\left(\sqrt{x}+1\right)\left(x+1\right)}\right).\frac{x+1}{\sqrt{x}+1}\)

\(=\frac{x\sqrt{x}+\sqrt{x}+1}{\left(\sqrt{x}+1\right)\left(x+1\right)}.\frac{x+1}{\sqrt{x}+1}\)

\(=\frac{x\sqrt{x}+\sqrt{x}+1}{x+2\sqrt{x}+1}\)

ko bik làm hay làm biến đó

3⁷⁵=(3³)²⁵=27²⁵

4⁵⁰=(4²)²⁵=16²⁵

vì 27²⁵>16²⁵=>3⁷⁵>4⁵⁰

vậy 3⁷⁵>4⁵⁰

Ta co :3⁷⁵ va 4⁵⁰

=> 3⁷⁵ = (3¹⁵)⁵=14348907

=>4⁵⁰ = (4¹⁰)⁵ = 1048576

Vi : 14348907 > 1048576

Vay 3⁷⁵ > 4⁵⁰