Effect of changing MC on monopolists' maximised profits
If MC increases, profits need to go down. Let $\pi (q(c))$ and $\pi' (q(c'))$ denote profits before and after the cost increase. Observe that producing $q(c')$ was feasible when the marginal cost was low and, therefore, it must be the case that

$$ \pi (q(c))=[p(q(c))-c]q(c)\geq[p(q(c'))-c]q(c').$$

In addition, we have that

$$[p(q(c'))-c]q(c')&gt;[p(q(c'))-c']q(c')=\pi' (q(c')),$$

which gives us $\pi (q(c))&gt;\pi' (q(c')).$