Science:Math Exam Resources/Courses/MATH104/December 2016/Question 06 (a)/Solution 1

Following the hint, we minimize the average cost $C_{\text{avg}}(q)={\frac {C(q)}{q}}$ by finding its critical points. The derivative of the average cost (with respect to quantity) is

$C_{\text{avg}}'(q)={\frac {C'(q)q-C(q)}{q^{2}}}$

by the quotient rule, which vanishes exactly when the numerator is zero. In other words, the critical points of $C_{\text{avg}}$ satisfy

$C'(q)={\frac {C(q)}{q}},$

which by definition means that the average cost is equal to the marginal cost.