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Math Help - The Derivative

  1. #1
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    The Derivative

    May someone please show me how to proof the following:

    Let f be differentiable on I. Show that if f is monotone increasing on I, then
    f '(x) >= 0 for all x in I.

    Note: >= is greater or equal to.

    My idea:
    Suppose that f is monotone increasing, that is f(x) <= f(c) for all x such that x<c. ... Then I know that after, we have to somehow say that [f(x)-f(c)]/(x-c) >= 0.
    Can someone explain more and do we need any theorem to support the proof?
    Thank you .
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  2. #2
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    Quote Originally Posted by zxcv View Post
    May someone please show me how to proof the following:

    Let f be differentiable on I. Show that if f is monotone increasing on I, then
    f '(x) >= 0 for all x in I.

    Note: >= is greater or equal to.

    My idea:
    Suppose that f is monotone increasing, that is f(x) <= f(c) for all x such that x<c. ... Then I know that after, we have to somehow say that [f(x)-f(c)]/(x-c) >= 0.
    Can someone explain more and do we need any theorem to support the proof?
    Thank you .
    Suppose that for some x \in I\ f'(x)<0.

    Let h>0 then as f is monotone increasing:

     <br />
\frac{f(x+h)-f(x-h)}{h} \ge 0<br />

    Hence:

     <br />
\lim_{h \to 0}\frac{f(x+h)-f(x-h)}{h} = f'(x) \ge 0<br />

    a contradiction, so there is no such point.

    CB
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    Hence:

    <br />
\lim_{h \to 0}\frac{f(x+h)-f(x-h)}{h} = f'(x) \ge 0<br />

    a contradiction, so there is no such point.
    Can you explain that a bit more. I'm getting lost in the letters.
    Last edited by HeirToPendragon; April 18th 2009 at 09:21 PM.
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  4. #4
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    Quote Originally Posted by HeirToPendragon View Post
    Can you explain that a bit more. I'm getting lost in the letters.
    If f'(x)<0 at some x point in I, then at that point (this uses the definition of having a derivative at that point):

    <br />
\lim_{h \to 0}\frac{f(x+h)-f(x-h)}{h} = f'(x) < 0<br />

    but we have just shown that as h approaches 0 from the left (using the assumption that f is monotone increasing and the existance of the derivative at the point):

    <br />
\lim_{h \to 0}\frac{f(x+h)-f(x-h)}{h} = f'(x) \ge 0<br />

    which is a contradiction.

    The only properties it uses are that f is monotone increasing and the definition of a derivative.

    CB
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  5. #5
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    I understand all that. What I'm confused on is how you know that f'(x) > 0

    Is there a theorem out there or a general rule I'm not understanding?

    We know that
    \frac{f(x+h)-f(x-h)}{h} \ge 0

    But how does that prove that
    \lim_{h \to 0}\frac{f(x+h)-f(x-h)}{h} = f'(x) \ge 0
    What is the reason that f'(x) can't be < 0 there?
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  6. #6
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    Quote Originally Posted by HeirToPendragon View Post
    I understand all that. What I'm confused on is how you know that f'(x) > 0

    Is there a theorem out there or a general rule I'm not understanding?

    We know that
    \frac{f(x+h)-f(x-h)}{h} \ge 0

    But how does that prove that
    \lim_{h \to 0}\frac{f(x+h)-f(x-h)}{h} = f'(x) \ge 0
    What is the reason that f'(x) can't be < 0 there?[/font]

    If f(x) is monotone increasing what is the sign of:

    \frac{f(x+h)-f(x-h)}{h}

    when h>0 ?

    CB
    Last edited by CaptainBlack; April 19th 2009 at 08:21 AM.
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  7. #7
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    What do you mean the sign?

    Do you mean is it positive or negative?

    It's positive. That makes perfect sense. But what I don't understand is why you know the lim of it is also positive.

    If a(x) is positive does that mean that it's lim has to be?
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  8. #8
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    Quote Originally Posted by HeirToPendragon View Post
    What do you mean the sign?

    Do you mean is it positive or negative?

    It's positive. That makes perfect sense. But what I don't understand is why you know the lim of it is also positive.

    If a(x) is positive does that mean that it's lim has to be?
    No it means the limit cannot be negative.

    CB
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  9. #9
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    Ok so what is the proof/reasoning for that?

    I don't mean to sound annoying, but I really want to know why I'm allowed to just say this.
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