Average Cost = Marginal Cost

[TheNextOne]

Let c = G(q) represent the total cost of distributing q > 0 litres of gas by an
oil company. It is assumed that G is differentiable and that the average cost has a relative extrema at a value q0. Use calculus to mathematically verify that, when q = q0, the average cost equals the marginal cost.

How would i do this?

The average cost would be G(q0)/(q0)
The Marginal Cost would be ????

 

[pka]

By definition, average cost is \(\displaystyle AC(q) = \frac{{C(q)}}{q}\). So marginal average cost is \(\displaystyle MAC(q) = \frac{{C'(q)q - C(q)}}{{q^2 }}\) the derivative of average cost.
In order for average cost to have an extreme
\(\displaystyle \begin{array}{l}
MAC(q_0 ) = 0\quad \Rightarrow \quad \frac{{C'(q_0 )q_0 - C(q_0 )}}{{q_0 ^2 }} = 0 \\
C'(q_0 )q_0 - C(q_0 ) = 0\quad \Rightarrow \quad C'(q_0 ) = \frac{{C(q_0 )}}{{q_0 }} \\
\end{array}\).

Or at q<SUB>0</SUB> marginal cost equals average cost.