Let $a,b,c,d,e,f$ be non negative real numbers. Find the minimum and maximum value of $ab+bc+cd+de+ef+fa$ given that $a+b+c+d+e+f=12$.
The minimum value will come as $0$ since we can take $a=b=c=d=e=0$ and $f=12$.
But how to find the maximum value? I tried to rearrange the terms but I couldn't be successful. I feel that the maximum value is $36$ since we can put $a=b=6$ and set the rest of the terms as $0$ and then the maximum value becomes $36$.
But these are just experimental verifications but can someone provide a strong, rigorous proof for both minimum and maximum value? Why can't we apply $AM-GM$ to get the minimum value?