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Math Help - How do you Use the Mean Value Theorem to prove this

  1. #1
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    How do you Use the Mean Value Theorem to prove this

    Use the Mean Value Theorem to prove that if p > 1 then (1+x)^p > 1+px for
    x \in (−1, 0) U (0, \infty).
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  2. #2
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    Quote Originally Posted by maximus101 View Post
    Use the Mean Value Theorem to prove that if p > 1 then (1+x)^p > 1+px for
    x \in (−1, 0) U (0, \infty).
    Dear maximus101,

    Using the Taylor's theorem you could write,

    (1+x)^p=1+px+p(p-1)(1+c)^{p-2}x^2~\text{where}~0<c<x~or~x<c<0------(A)

    p>1\Rightarrow{p(p-1)>0}-----(1)

    If 0<c<x~then~1<1+c-----(2)

    If x<c<0~then~{1+x<1+c<1}----(3)

    x>-1\Rightarrow{1+x>0}------(4)

    By, (3) and (4); If~x<c<0\Rightarrow{0<1+c<1}----(5)

    By (2) and (5); for both cases, 0<c<x~and~x<c<0\Rightarrow{0<1+c}------(6)

    Therefore, by (1) and (6); p(p-1)(1+c)^{p-2}x^2>0~if~x\neq{0}~and~x>-1-----(6)

    By (A);

    (1+x)^p-1-px=p(p-1)(1+c)^{p-2}x^2>0~if~x\neq{0}~and~x>-1

    (1+x)^p>1+px~if~x>-1~and~x\neq{0}
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  3. #3
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    Hey, thank you for this it was very helpful could you explain how I could use the mean value theorem to prove it?
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    Quote Originally Posted by maximus101 View Post
    Hey, thank you for this it was very helpful could you explain how I could use the mean value theorem to prove it?
    There are several mean value theorems; Rolle's mean value theorem, Cauchy's mean value theorem and the Taylor's mean value theorem. Please refer Mean value theorem - Wikipedia, the free encyclopedia. I have used the Taylor's mean value theorem in the above answer.
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    Quote Originally Posted by Sudharaka View Post
    There are several mean value theorems; Rolle's mean value theorem, Cauchy's mean value theorem and the Taylor's mean value theorem. Please refer Mean value theorem - Wikipedia, the free encyclopedia. I have used the Taylor's mean value theorem in the above answer.
    Hey, sorry for my mistake, I meant the first one showing

    f'(c)= \frac{f(b)-f(a)}{b-a}

    kind thanks
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  6. #6
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    Quote Originally Posted by maximus101 View Post
    Hey, sorry for my mistake, I meant the first one showing

    f'(c)= \frac{f(b)-f(a)}{b-a}

    kind thanks
    That is also know as Lagrange's mean value theorem. It is used to prove Taylor's theorem. Therefore it had been used in the answer.
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  7. #7
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    ok I worked it out thank you
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