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Math Help - Prove using the mean value theorem

  1. #1
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    Prove using the mean value theorem

    Using mean value theorem , prove that
    tan x > x for all x \boldsymbol{\epsilon} ( 0, \frac{\pi}{2} )
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  2. #2
    MHF Contributor Calculus26's Avatar
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    For all x in (0,pi/2) there is a c in (0,x) such that

    [tan(x)-tan(0)]/[x-0] = sec^2(c) > 1 for c in (0,pi/2)

    tan(x)/x > 1

    tan(x) > x
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  3. #3
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    I did get this part

    Quote Originally Posted by Calculus26 View Post
    For all x in (0,pi/2) there is a c in (0,x) such that

    [tan(x)-tan(0)]/[x-0] = sec^2(c) > 1 for c in (0,pi/2)

    tan(x)/x > 1

    tan(x) > x

    I didnt get this part could u please explain

    [tan(x)-tan(0)]/[x-0] = sec^2(c)
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  4. #4
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    Quote Originally Posted by zorro View Post
    I didnt get this part could u please explain

    [tan(x)-tan(0)]/[x-0] = sec^2(c)
    that is just applying the mean value theorem. [f(b)-f(a)] / (b-a) = f'(c), in this case a=0, b=x, and f(x) = tan(x) so f'(x) = sec^2(x).
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