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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.

enter image description here

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?

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I assume $R(t)$ is cumulative. If it isn't, you have one heck of a success story in that graph! Actually, the profit you make is $$ P(t) = R(t) - 10,000t $$ Since every extra day costs 10,000. This function DOES have a maximum where its derivative is equal to 0. So, you want $$ P'(t) = R'(t) - 10,000 = 0 \implies R'(t) = 10,000 $$ You need to figure out approximately where your graph has slope 10,000 and that is your point. My estimate on your graph places that point between 20 and 25 days.

Hope this helps!

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