Let $\pi(y) = R(y) - C(y)$ be profits where $R(y)$ is revenue and $C(y)$ is costs. Let $R(y) = p_y y$. Then
\begin{align*} \frac{\partial \pi }{\partial y} &= 0\\ \frac{\partial x}{\partial y}(p_y y -C(y)) &= 0\\ C'(y) &= p_y\\ \end{align*}
Thus, $MR = MC$ in this case.
Now define $r(y) = \frac{\pi(y)}{C(y)}$ to be the return. Intuitively, suppose you own the business. $C(y)$ is the amount you put in and $\pi(y)$ is what you get back. You want to maximize your return.
\begin{align*} \frac{\partial r}{\partial y} &=0\\ \frac{\partial}{\partial y} \frac{p_y y -C(y)}{C(y)} &=0\\ \frac{C(y)p_y - p_y y C'(y)}{[C(y)]^2} &=0\\ C(y)p_y - p_y y C'(y) &=0\\ C'(y) &=\frac{C(y)}{y}\\ \end{align*}
Thus, $MC = ATC$.
But when I googled this, I unexpectedly found this passage:
The point at which marginal cost equals average total cost (MC = ATC) is known as the break-even point. When the MR = P line crosses through this point, as is highlighted by the black circle on the graph, the product is said to be selling at its break-even price because the marginal revenue will exactly offset the marginal cost of production, and total revenue will exactly offset total cost. In this situation, the firm will break even: it will not be earning any profits, but it will not be losing money either. If the MR = P line lies above the break-even point, the firm will be operating at a profit, since the revenue earned on each unit of output sold will exceed the average cost of producing a unit of output, and thus total revenue will exceed total cost. If the MR = P line lies below the break-even point, the firm will be operating at a loss because the revenue earned on each unit of output will be less than the average cost of producing a unit of output, and so total revenue will be less that total cost.
I don't understand why, if you are maximizing the return for a founder investing in their business, why they would be getting the best return when they are breaking even. That doesn't make sense to me.
Can someone explain why $MC = ATC$ is both the point where an investor would be getting their max return and the breakeven point?