I have that, $\forall \theta^\prime$ $$u(\theta)=\max_{\theta^{\prime }\in[\theta_{1}, \theta_{2}]}(\theta q(\theta^\prime)-t(\theta^\prime))$$
Given any value $\theta^\prime, \theta q(\theta^\prime)-t(\theta^\prime)$ is an increasing and linear function of $\theta$. I am told that $$\lim_{\delta\rightarrow0}\frac{u(\theta+\delta)-u(\theta)}{\delta}\geq\lim_{\delta\ \rightarrow0}\frac{((\theta+\delta)q(\theta)-t(\theta))-(\theta q(\theta)-t(\theta))}{\delta}$$ and $$\lim_{\delta\rightarrow0}\frac{u(\theta)-u(\theta-\delta)}{\delta}\leq\lim_{\delta\ \rightarrow0}\frac{(\theta q(\theta)-t(\theta))-((\theta-\delta)q(\theta)-t(\theta))}{\delta}$$ Why is this the case? I understand that both left hand side terms define $u^\prime(\theta)$, while both right hand side terms define $\frac{d(\theta q(\theta)-t(\theta))}{d\theta}.$ But why does the inequality become ‘flipped’ in the second line? Thanks.