<s>[INST] Prove that $f(a,c,b,d) > f(a,b,c,d) > f(a,b,d,c).$ 
Let $f(a,b,c,d) = (a-b)^2+(b-c)^2+(c-d)^2+(d-a)^2$. For real numbers $a < b < c < d,$ prove that $f(a,c,b,d) > f(a,b,c,d) > f(a,b,d,c).$

Would the best way to do this be to expand each expression and show directly the inequality which we need to prove?
 [/INST] 
Question
Let $f(a,b,c,d) = (a-b)^2+(b-c)^2+(c-d)^2+(d-a)^2$. 
Prove that $f(a,c,b,d) > f(a,b,c,d) > f(a,b,d,c), $ if  $a < b < c < d$. 

Then 
$$\begin{align}
f(a,c,b,d)&=(a-c)^2+(c-b)^2+(b-d)^2+(d-a)^2\\
&=(a-c)^2+(b-c)^2+(b-d)^2+(d-a)^2 
\end{align}$$
Now 
$$\begin{align}
f(a,c,b,d) - f(a,b,c,d)&=(a-c)^2+(b-d)^2-((a-b)^2+(c-d)^2)\\
&=2ab+2cd-2ac-2bd \\
&= 2a(b-c)-2d(b-c)\\
&=2(a-d)(b-c) >0.
\end{align}
$$
(Why is that inequality valid?)
This method works for the second inequality as well.
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