After looking at this post, I framed the following question.
If $a,b,c,d,e,f$ are positive real numbers such that $a+b+c+d+e+f=1$, then find the minimum and maximum value of $ab+bc+cd+de+ef+fa$
My Attempt:
$\frac{a+b+c+d+e+f}6\ge(abcdef)^{\frac16}\implies\frac1{36}\ge(abcdef)^{\frac13}$
Also, $\frac{ab+bc+cd+de+ef+fa}6\ge(abcdef)^{\frac13}$
Dividing the second inequality by first, I get,
$ab+bc+cd+de+ef+fa\ge\frac16$
I got the minimum value. For maximum value, I think this post has a solution, but not able to understand how it framed the inequalities.
Also, the minimum value could have been achieved by assigning each variable as $\frac16$. In such equality cases, how do we know whether we'll get minimum value or maximum value, without solving algebraically?