Maximium value of $(b-a)\Big(\dfrac 34-\dfrac{a+b}2-\dfrac{a^2+ab+b^2}3\Big)$ For $b>a$ what is the maximum possible value of $(b-a)\Big(\dfrac 34-\dfrac{a+b}2-\dfrac{a^2+ab+b^2}3\Big)$ ? 

For a fixed $a$, the expression is
$$\frac34(b-a)-\frac12(b^2-a^2)-\frac13(b^3-a^3)$$
which is a third degree polynomial.
Differentianting respect to $b$ gives:
$$\frac 34-b-b^2$$ whose roots are $-3/2$ and $1/2$.
The maximum is reached when $b=1/2$. If $a>1/2$, there is no maximum.
