Consider a general cost function C(q), with Average Cost=C(q)/q. Prove

Consider a general cost function C(q), with Average Cost=C(q)/q. Prove using calculus that the quantity which minimizes the Average Cost function is also at the point where the Marginal Cost equals the Average Cost.

To do this, first derive the necessary condition for the AC to be at a minimum.

ANSWER

Consider a cost function C(q). Then AC(q) = C(q)/q. In order for AC to be at a minimum, the following condition must hold:
dAC(q)/dq = 0
Using the division rule for differentiation,
dAC(q)/dq = d(C(q)/q)/dq = (qC'(q) – C(q))/q = 0
The numerator must equal zero for the left hand side to equal zero (q>0).
qC'(q) – C(q) = 0
Rearrange and using the fact that C'(q) = MC(q)
MC(q) = AC(q)