Finding Maximum Under Constraint Suppose $a$,$b$,$c$ satisfy $a+b+c=1$ and $a$,$b$,$c\in [0,1]$
Find the maximum value of $(a-b)(b-c)(c-a)$

WOLOG, suppose $c$ is the largest. If $a>b$, then $(a-b)(b-c)(c-a)\le 0$. So we may suppose $a<b<c$. Let $b-a=x\ge 0, c-b=y\ge 0$, then $3a+2x+y=1$, thus $2x+y\le 1$. The problem becomes find maximum value of 
$$
xy(x+y), \quad x, y \ge0.
$$ 
So we may suppose $2x+y=1$, then
$$
xy(x+y)=x(1-2x)(1-x)=:f(x), \quad 0\le x\le \frac{1}{2}.
$$
Put $f'(x)=1-6x+6x^2=0$, when $x=\frac{3-\sqrt{3}}{6}$, $f(x)$ attains the maximum.
