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# Help!

0
104
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+1206

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
+1

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

Jun 9, 2019
#2
+103674
+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