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

 Jun 9, 2019
 #2
avatar+102466 
+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

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