<s>[INST] finding minimum of function Can you please give me some hints finding minimum of this function:
$ (r-1)^2 + (\frac{s}{r} -1)^2 + (\frac{t}{s}-1)^2 + (\frac{4}{t}-1)^2$  
where $ 1 \le r \le s \le t \le 4 $,
$r,s,t \in \Bbb R $
 [/INST] Another way...  As Daniel Fischer has done, define $a=r, b=\frac{s}r, c=\frac{t}s, d=\frac4t$.  Then we have $a, b, c, d \ge 1$ and $abcd = 4$.  
Now using the QM- AM-GM inequalities, we have
$$\sqrt{\frac{(a-1)^2 + (b-1)^2 + (c-1)^2+(d-1)^2}{4}}\ge \frac{a+b+c+d - 4}4 \ge \sqrt[4]{abcd}-1 = \sqrt{2}-1$$
with equality (i.e. minimum) iff $a=b=c=d=\sqrt2$.
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