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Math Help - Differentiable function problem

  1. #1
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    Differentiable function problem

     f is a differentiable function at  [a,\infty).
    i need to prove that:
    if there is a constant m>0 which maintains that f'(x)\geq m to every x\in[a,\infty),
    so  lim_{x\rightarrow\infty}f(x)=\infty ... thx!!
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  2. #2
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    Re: Differentiable function problem

    Quote Originally Posted by orir View Post
     f is a differentiable function on  [a,\infty).
    i need to prove that:
    if there is a constant m>0 which maintains that f'(x)\geq m to every x\in[a,\infty),
    so  lim_{x\rightarrow\infty}f(x)=\infty ... thx!!

    If x>a then \exists x'\in(a,x) such that \frac{f(x)-f(a)}{x-a}=f'(x')\ge m. WHY?

    That means that f(x)\ge m(x-a)+f(a). What can you do with that?
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  3. #3
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    Re: Differentiable function problem

    Quote Originally Posted by Plato View Post
    If x>a then \exists x'\in(a,x) such that \frac{f(x)-f(a)}{x-a}=f'(x')\ge m. WHY?

    That means that f(x)\ge m(x-a)+f(a). What can you do with that?
    can i say that \varepsilon=m(x-a)+f(a)? so then,  f(x)\ge\varepsilon??
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  4. #4
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    Re: Differentiable function problem

    Quote Originally Posted by orir View Post
    can i say that \varepsilon=m(x-a)+f(a)? so then,  f(x)\ge\varepsilon??
    Well, you did not answer the question WHY?

    If you want further help, then you must explain where the inequality comes from.

    Moreover, what does it mean to say that {\lim _{x \to \infty }}f(x) = \infty~? Give the precise definition.
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  5. #5
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    Re: Differentiable function problem

    oh, i know why. it's because the mean value theorem.
    and.. the precise definition is that there is an m>0, and n>0, so that for every x larger than m, f(x) larger than n.
    Last edited by orir; May 8th 2013 at 09:10 AM.
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    Re: Differentiable function problem

    Quote Originally Posted by orir View Post
    oh, i know why. it's because the mean value theorem.
    and.. the precise definition is that there is an m>0, and n>0, so that for every x larger than m, f(x) larger than n.

    Can you show that {\lim _{x \to \infty }}\left( {m(x - a) + f(a)} \right) = \infty ~?

    If so, what does that imply?
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  7. #7
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    Re: Differentiable function problem

    i guess i can show that. but i don't know what does that imply...
    and what about my way? to say that  n=m(x - a) + f(a) and that way show the right of the definition above.
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  8. #8
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    Re: Differentiable function problem

    Quote Originally Posted by orir View Post
    i guess i can show that. but i don't know what does that imply...
    and what about my way? to say that  n=m(x - a) + f(a) and that way show the right of the definition above.

    Well \forall x>a we have shown that f(x)\ge m(x-a)+f(a)\to\infty~.
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  9. #9
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    Re: Differentiable function problem

    oh, i got you..
    but still - does my way work too?
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