f and g are cont on [0,1], then f and g are integrable.

Let

Let and

then both are continuous on [0,1], which implies that is differentiable on [0,1], and , for all (by a corollary to the FTOC)

. qed

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- Nov 10th 2007, 07:12 PM #1

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## Problem 40

1)This problem is for the younger kids give them a chance, please. Let be randomly chosen from (they can be the same). What is the probability that it is possible to find two

*real*numbers that have their product equal to and their sum equal to . (By the way, "real", means non-imaginary).

2)Let be continous on so that for all . Prove that . (This is a classic).

- Nov 10th 2007, 08:03 PM #2

- Nov 10th 2007, 08:16 PM #3

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I am sorry, I cannot follow.

How is this a well-defined function? Maybe for a particular value of ?

then both are continuous on [0,1], which implies that is differentiable on [0,1], and , for all (by a corollary to the FTOC)

---

Remember I am saying that:

, , , ...

- Nov 10th 2007, 11:29 PM #4

- Nov 10th 2007, 11:37 PM #5

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- Nov 10th 2007, 11:46 PM #6
Hmm, that is much simpler than the method I thought of in my head (I'm a bit too embarrassed to say what my method was, but I'll just say it involves a new pack of pencils, a full ream of paper, a gallon of coffee, a carton of cigarettes, and no obligations for several days.)

- Nov 10th 2007, 11:50 PM #7

- Nov 11th 2007, 12:10 AM #8
i always have trouble proving things that seem to me to be obvious.

would it be wrong to do this?

Obviously, the only way this integral can be zero for all x and all n is if . the result follows immediately.

Now i can see where the problem here would be, the "obviously" part.

- Nov 11th 2007, 12:49 AM #9

- Nov 11th 2007, 05:23 AM #10

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- Nov 11th 2007, 08:07 AM #11

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- Nov 11th 2007, 09:32 AM #12

- Nov 11th 2007, 11:34 AM #13

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- Nov 11th 2007, 11:50 AM #14

- Nov 11th 2007, 11:56 AM #15

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