Here are some especially tricky and/or fun problems. I'll post them at random intervals as we develop the required tools or as I find them. These are not required (unless I specify in class that you should at least think about one of them) but good solutions may well earn rewards. Most rewards will be chocolate-based.

Let f be an infinitely differentiable function from to . Suppose that, for some positive integer n ,

f(1) = f(0) = f'(0) = f''(0) = ⋅⋅⋅ = f⁽ⁿ⁾(0) = 0.

Prove that f^{(n + 1)}(x) = 0 for some x∈(0,1) .

A more readable version is attached below as "chall1.pdf". That document also contains some additional notes.

http://cs.marlboro.edu/ courses/ fall2008/analysis/ challenge
last modified Thursday September 11 2008 11:45 am EDT

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chall1.pdf Sep 8 2008 11:02 am 37.6kB