I have the following problem:
You want to put up a billboard and the ad costs $10,000 per day. Your market research team has produced the graph below. In it, R(t) predicts the extra sales, in hundreds of thousands of dollars, that a posting of length t days will produce. You can buy an of any length days between 0 and 30 days. Decide how long the board should should be up.
I don't know how to do this, except to pick some points, try to guess the value of R(t) for it, and see the profit. But this is ad-hoc. And the derivative never looks like it goes to 0 so the normal optimization tactic fails (though normally I'd do that with a formula too, but I can't see a spot where it would be 0). Is there a better way to do this without knowing what R(t) is?