In today's test (question c) I had to minimize equation $(3)$ and solve for N*.
I did it through deriving, setting to $0$ and solve for N (no doubts about that).
My question is, in this image it looks misleading to me (and other students), when they have the minimum point directly above the intersection of the two components of the total cost function (implied by the dashed line). Is it actually possible to find N by calculating the intersection point?
This is true when the slopes of the two components are negatives of each other at the intersection point AND the functions are monotonic. This is not always true.
However, for equations of the form $f(x) = \frac{A}{x} + Bx$, such as this example, it is.
Setting derivative to zero: $-\frac{A}{x^2} + B = 0$
Setting components equal: $\frac{A}{x} = Bx$
We see that they both have the same solution. However, this is somewhat coincidental due to the nature of the function, so the first method is preferable.