<s>[INST] Minimum without computing the derivative. I am trying to find the minimum of the following function:
$$H(x)=\frac{(x+\frac{1}{x})^6-(x^6+\frac{1}{x^6})-2}{(x+\frac{1}{x})^3+(x^3+\frac{1}{x^3})},\hspace{1cm}x>0$$
What I usually do to find extrema of a function is computing the derivative of the function and finding its zeros. However, I think there must be a more efficient approach to this problem, since the derivative of this function is quite long. Any tips?
Thanks.
 [/INST] $$\displaystyle \bigg[\bigg(x+\frac{1}{x}\bigg)^3\bigg]^2-\bigg[x^3+\frac{1}{x^3}\bigg]^2$$
$$\bigg[\bigg(x+\frac{1}{x}\bigg)^3+x^3+\frac{1}{x^3}\bigg]\bigg[\bigg(x+\frac{1}{x}\bigg)^3-x^3-\frac{1}{x^3}\bigg]$$
$$H(x)=\bigg(x+\frac{1}{x}\bigg)^3-x^3-\frac{1}{x^3}=3\bigg(x+\frac{1}{x}\bigg)\geq 3\cdot 2=6$$
Because arithmetic geometric inequality
$$x+\frac{1}{x}\geq 2\;\forall \;x>0$$
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