+1245 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.
+118673 \(\frac{\left( x + \dfrac{1}{x} \right)^6 - \left( x^6 + \dfrac{1}{x^6} \right) - 2}{\left( x + \dfrac{1}{x} \right)^3 + \left( x^3 + \dfrac{1}{x^3} \right)} \)
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,
