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Math Help - Analysis differentiation

  1. #1
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    Analysis differentiation

    Hi, I can't seem to get anywhere with the following problems:

    1. Find a differentiable function f: R->R such that f'(0)=1 but f is not monotonically increasing on any interval (0,d).

    2. Show that there exists a differentiable function f: R->R such that (f(x))^5 + f(x) + x = 0 for all x. [Hint: if f exists and has an inverse g what equations must g satisfy?]

    Any help would be much appreciated.

    xxxxxx
    Last edited by robward69; February 25th 2009 at 03:25 PM.
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  2. #2
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    1. Find a differentiable function f: R->R such that f'(0)=1 but f is not monotonically increasing on any interval (0,d).
    How about the function f(x)=x+x^2\sin\frac\pi x for x\neq0, f(0)=0 ? Clearly the function is differentiable at all non-zero x, whereas

    \left|\frac{f(h)-f(0)}h-1\right|\leq|h| for h\neq0, making f'(0)=1.

    Furthermore if x\neq0 then f'(x)=1+2x\sin\frac\pi x-\pi\cos\frac\pi x.

    Thus f'(1/n)=1-\pi(-1)^n for all positive integers n, showing that the function is not monotonic on any interval (0,a).

    Of course, the function f'(x) cannot be continuous at 0 if it is to meet the conditions of the problem, wouldn't you say?
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