Every man for himself and the devil take the hindmost
It is 1636 and you are interested in buying tulips as an investment. As part of your research, you monitor the price of tulips for ten consecutive weeks and find that the price increases, each week by a greater amount than it did in the last. You decide that the price of tulips is on an upwards trend. Therefore, you can buy tulips now and sell them for more money before the price goes down. Just before you place the trade, a time traveller who is well versed in calculus, appears and gives you advice on the trade. What do they tell you about your plan?
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1 solution
This problem makes the assumption that the price increase each week is independent of the price in the previous week (which is not 100% accurate but makes some intuitive sense). If the price today is independent from the price yesterday, there is no way to predict given yesterday's price what today's will be.
Appealing to the tags given (one of which is #IntermediateValueTheorem), I don't see much of a straightforward way to use the IVT to show that current trends are independent of the future ones except for the fact that you cannot "bind" future values into a range without first having a closed range [ a , b ] with corresponding y-values; unfortunately, this cannot be extended into the future as future prices aren't known.