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Derivatives and deltas.

  1. May 28, 2012 #1
    1. The problem statement, all variables and given/known data
    Hi

    I've been giving the following problem:
    We have a differentiable function f: [a,b] [itex]\rightarrow[/itex] [itex]\mathbb{R}[/itex] with f'(a) < 0 en f'(b) > 0. Let c [itex]\in[/itex] [itex]\mathbb{R}[/itex] such that f'(a) < c. Show that there exists a [itex]\delta[/itex] >0 such that for every x [itex]\in[/itex] ]a, a + [itex]\delta[/itex][ the following holds:

    f(x) < f(a) + c(x-a).
    2. Relevant equations



    3. The attempt at a solution
    My attempt at a solution is the following:
    Using the definition of the derivative we have the following:

    lim x [itex]\rightarrow[/itex] a f(x) - f(a)/(x - a) < c so f(x) < f(a) + c(x-a).

    My question is, where do I get the interval ]a, a + [itex]\delta[/itex][ from?
     
  2. jcsd
  3. May 28, 2012 #2

    LCKurtz

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    Re: Derivative's and delta's.

    You have $$
    \frac {f(x) - f(a)}{x-a}\rightarrow f'(a)<c$$ as ##x\rightarrow a##. That doesn't mean it is less than ##c## for all ##x##. ##x## has to be sufficiently close to ##a##. You need to think about the definition of limit to get a ##\delta## that works.
     
  4. May 28, 2012 #3
    Re: Derivative's and delta's.

    Okay I've figured it out. Thanks for your help ^^.
     
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