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 $(b-a)\Big(\dfrac 34-\dfrac{a+b}2-\dfrac{a^2+ab+b^2}3\Big)$ ? 

By partial differentiation:
Assume $b$ fixed, differentiate with respect to $a$, we get the condition
$$ a^2 + a - \frac {3}{4} = 0 $$
Assume $a$ fixed, differentiate with respect to $b$, we get the condition
$$ b^2 -b - \frac {3}{4} = 0 $$
This gives us $(a, b) = ( - \frac{3}{2}, \frac{1}{2} ) $. It remains to verify if this is indeed a local maximum, and  if it is the global maximum. This gives us the value $ \frac{4}{3} $.
