<s>[INST] the maximum value of $(a+\frac1b)(b+\frac1c)(c+\frac1d)(d+\frac1a)$ 
If $a, b, c, d \in [\frac12, 2]$ and $abcd =1$, find the maximum value of 
  $$(a+\frac1b)(b+\frac1c)(c+\frac1d)(d+\frac1a)$$

Thought: $$\begin{split}(a+\frac1b)(b+\frac1c)(c+\frac1d)(d+\frac1a) &= \frac{(ab+1)(bc+1)(cd+1)(da+1)}{abcd} \\ & \stackrel{\text{C-S}}{\le} \sqrt{(a^2+1)(b^2+1)}\sqrt{(b^2+1)(c^2+1)}\sqrt{(c^2+1)(d^2+1)}\sqrt{(d^2+1)(a^2+1)} \\ & = (a^2+1)(b^2+1)(c^2+1)(d^2+1)\end{split}$$
 [/INST] I think it can help:
http://www.artofproblemsolving.com/community/c6h621917
See an elim's proof.
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