Decreasing Costs, Increasing Returns to Scale, & C''(q)
Second derivative of cost function is actually the first derivative of marginal cost function. i.e. 
$$
\frac{\partial^2C(q)}{\partial q^2} =\frac{\partial}{\partial q}\frac{\partial C(q)}{\partial q}=\frac{\partial}{\partial q}MC(q)
$$
Now if $\frac{\partial^2C(q)}{\partial q^2}&lt;0$, this means that marginal cost is decreasing in output. If marginal cost is decreasing then that implies that firm's average cost is decreasing and hence it exhibits increasing returns to scale.