+0  
 
0
291
2
avatar+1211 

Find the minimum value of \(\frac{\left( x + \dfrac{1}{x} \right)^6 - \left( x^6 + \dfrac{1}{x^6} \right) - 2}{\left( x + \dfrac{1}{x} \right)^3 + \left( x^3 + \dfrac{1}{x^3} \right)}\) for x > 0. 

 Jun 9, 2019
 #1
avatar
+1

The expression has a minimum value of 6, when x = 1.

 Jun 9, 2019
 #2
avatar+109724 
+1

\(\frac{\left( x + \dfrac{1}{x} \right)^6 - \left( x^6 + \dfrac{1}{x^6} \right) - 2}{\left( x + \dfrac{1}{x} \right)^3 + \left( x^3 + \dfrac{1}{x^3} \right)} \)

 

Just by inspection I can see that as x moves away from 1 in either direction the function value will increase.

So the minimum will occur when x=1

that minimum will be

 

\(\frac{2^6-2-2}{2^3+2}=\frac{60}{10}=6\)

 

Just as our guest said,

 Jun 10, 2019

9 Online Users

avatar
avatar
avatar
avatar