There is a class of problems that I noticed comes up again and again in various scenarios. Abstractly, you can formulate it like this: given a time limit, how much time should you spend increasing your production capacity, and then how much time should you use your production capacity to produce utility? Let's take a look at two version of this problem:
You have N days. You start with a production capacity C=0 and accumulated utility U=0. Each day you can either: 1) increase your production capacity (C=C+1) or 2) use your current production capacity to produce utility (U=U+C).
Question: On what days should you increase your production, and on what days should you produce utility to maximize total accumulated utility at the end of the N days?
It's trivial to prove that the optimal solution looks like increasing capacity for T days, and then switching to producing utility for N-T days. What is T? In this case it's really straight-forward to figure it out. We can compute final utility as U(T)=(N-T)*T. The maximum is at T=N/2. So, you should spend the first half increasing your production and the second half producing utility. Interesting...
You have N days. You start with a production capacity C=1 and accumulated utility U=0. Each day you can either: 1) increase your production capacity by a factor F (C=C*F), where F>1 or 2) use your current production capacity to produce utility (U=U+C).
Same question. Now the final utility is U(T)=F^T*(N-T). Doing basic calculus, we find the optimal T=max(0, N-1/ln(F)). A few interesting points you can take a way from this solution:
1) If your growth factor F is not large enough, you might have to stick with your original production capacity of 1 and never increase it. E.g. F=1.01 and N=100, where optimal T=max(0,-0.499171).
2) The bigger the N, the lower growth factor you can accept as being useful, i.e. T>0.
Questions for you: where have you seen these types of problems come up in your life? Is this a known class of problems?