From page 173, CS+PS=integral from 0 to q of p(x)-c so form the Lagrangian CS+PS-lambda.(p(q)q-cq-F). Differentiate wrt lambda to get p*-c=F/q>0, then differentiate wrt q to get the main result. (You will need the result that d/da(integral 0 to a of f(x)) equals f(a).)
Do not be put off by the fact that the Lagrangian multiplier is still in your formula - this is usual in Ramsey rules.