Proof that for increasing AC MC(Q)>AC(Q) at any Q>0? I know the "common sense" proof, but how can we prove it algebraically?
Let $C(y)$ denote the cost function and $y$ the quantity.

The average cost function is $AC=C/y$. This is increasing if its derivative is positive. Let $C'$ denote the derivative of $C$. The marginal cost function is $MC=C'$. We have:

$\frac{\partial AC}{\partial y}= \frac{C'y - C}{y^2}&gt;0$ by using the quotient rule. As $y^2&gt;0$, the condition becomes:

$C'y - C&gt;0$. 

Rearranging yields:

$C' &gt; C/y$ 

Hence: $MC&gt;AC$.