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Thread: 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; Feb 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 $\displaystyle f(x)=x+x^2\sin\frac\pi x$ for $\displaystyle x\neq0$, $\displaystyle f(0)=0$ ? Clearly the function is differentiable at all non-zero $\displaystyle x$, whereas

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

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

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

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