Monopolistic Competition- productive efficiency In my textbook, it says that monopolistic competition will never lead to productive efficiency (where $MC=AC$) in the short run and long run. I understand that they are meant to be producing at the profit-maximising level of output (where $MC=MR$), but all the diagrams showcase scenarios where the $AC$ curve is always positioned such that the points where MR=MC does not intersect where $MC=AC$.

My question is, is there any possibility that the profit-maximising level of output will also happen to be where we've achieved the lowest possible unit cost (lowest $AC$)? So that $MC=MR=AC$, and the firm can be both productively-efficient whilst also being profit-maximising? Why is this not possible?
Let $q$ denote the output of the firm, and let $\varepsilon_p(q)$ denote the elasticity of price w.r.t. quantity sold. We know that when profit is maximized
$$
|\varepsilon_p(q)| = \frac{p(q)-MC(q)}{p(q)}.
$$
We also know that in the long-run equilibrium firms have zero economic profit, i.e.
$$
AC(q) = p(q).
$$
Productive efficiency is achieved when $AC(q)$ is minimized, and at that point
$$
AC(q) = MC(q)
$$
holds as well. Combining these we get
$$
|\varepsilon_p(q)| = \frac{p(q)-MC(q)}{p(q)} = \frac{AC(q)-AC(q)}{p(q)} = 0.
$$
So the only case when all these properties coincide is when the quantity sold has no effect on the market price, i.e. when demand is perfectly elastic.



An explanation without the elasticity part: 
From the other two properties (long-run equilibrium and productive efficiency) we have 
$$
MC(q) = AC(q) = p(q).
$$
Together with profit maximization this implies that firms are price takers, $MR(q) = p(q)$, but since
$$
MR(q) = \frac{\text{d} \ p(q)q}{\text{d} \ q} = p(q) + p'(q)q
$$
this is only possible if $p'(q) = 0$, which again means that demand is perfectly elastic.



All the above assumed that the demand and inverse demand functions are continuously differentiable. You could have a situation with not perfectly elastic but kinked demand, where profit maximization occurs exactly at the kink. In this case $MR(q)$ does not exist at the point of profit maximization.