Minimal value problem | RMO 2015 Chennai Solution

This is a problem from Regional Mathematics Olympiad, RMO 2015 Chennai Region based on the Minimal value problem. Try to solve it.

Problem: Minimal value problem

Find the minimum value of $ \displaystyle { \frac{ ( x + \frac{1}{x} )^6 - ( x^6 + \frac{1}{x^6}) - 2}{(x+\frac{1}{x})^3 + (x^3 + \frac{1}{x^3} )} } $ and $ s=2$ and $ x \in \mathbf{R} $ and $ s=2 $ and $ x > 0 $ and $ s=2 $

Discussion:

$ \displaystyle { \frac{ ( x + \frac{1}{x} )^6 - ( x^6 + \frac{1}{x^6}) - 2}{(x+\frac{1}{x})^3 + (x^3 + \frac{1}{x^3} )} } $ and $ s=2$

$ = \displaystyle { \frac{ ( x + \frac{1}{x} )^6 - ( (x^3)^2 + (\frac{1}{x^3})^2) - 2}{(x+\frac{1}{x})^3 + (x^3 + \frac{1}{x^3} )} } $ and $ s=2$

$ = \displaystyle { \frac{ ( x + \frac{1}{x} )^6 - ( (x^3)^2 + (\frac{1}{x^3})^2 + 2)}{(x+\frac{1}{x})^3 + (x^3 + \frac{1}{x^3} )} } $ and $ s=2$

$ = \displaystyle { \frac{ ( x + \frac{1}{x} )^6 - ( (x^3)^2 + (\frac{1}{x^3})^2 + 2 \times (x^3) \times \frac{1}{x^3})}{(x+\frac{1}{x})^3 + (x^3 + \frac{1}{x^3} )} } $ and $ s=2$

$ = \displaystyle { \frac{ {( x + \frac{1}{x} )^3}^2 - (x^3 + \frac{1}{x^3})^2}{(x+\frac{1}{x})^3 + (x^3 + \frac{1}{x^3} )} } $ and $ s=2$

$ = \displaystyle { \frac{ {( x + \frac{1}{x} )^3 + (x^3 + \frac{1}{x^3})} {( x + \frac{1}{x} )^3 - (x^3 + \frac{1}{x^3})}}{(x+\frac{1}{x})^3 + (x^3 + \frac{1}{x^3} )} } $ and $ s=2$

$ = \displaystyle { ( x + \frac{1}{x} )^3 - (x^3 + \frac{1}{x^3})} $ and $ s=2$

$ = \displaystyle { 3( x + \frac{1}{x} )} $ and $ s=2$

Applying Arithmetic Mean - Geometric Mean Inequality (since x is positive), we have:

$ \displaystyle { \frac {( x + \frac{1}{x} )}{2} \ge \sqrt {x\times \frac{1}{x}} = 1 } $ and $ s=2$

$ \displaystyle { \Rightarrow ( x + \frac{1}{x} ) \ge 2 } $ and $ s=2$

$ \displaystyle { \Rightarrow 3( x + \frac{1}{x} ) \ge 3\times 2 = 6} $ and $ s=2$

Hence the minimum value is 6.

Chatuspathi:

• Paper: RMO 2015 (Chennai)
• What is this topic: Inequality
• What are some of the associated concepts: Arithmetic and geometric Inequality
• Where can learn these topics: Cheenta I.S.I. & C.M.I. course,Cheenta Math Olympiad Program, discuss these topics in the ‘Inequality’ module.
• Book Suggestions: Inequality by Little Mathematical Library, Secrets of Inequality