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Math Help - mean value theorem

  1. #1
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    mean value theorem

    Use the Mean Value Theorem to prove that sqrt(1+x) < 1 + x/2 for all x > 0. Generalize this result to the function (1+x)^r, where r < 1 is a positive rational number.

    Thanks!
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by friday616 View Post
    Use the Mean Value Theorem to prove that sqrt(1+x) < 1 + x/2 for all x > 0. Generalize this result to the function (1+x)^r, where r < 1 is a positive rational number.

    Thanks!
    Using the MVT, \frac{f(x)-f(0)}{x-0}=\frac{\sqrt{1+x}-1}{x}=f'(t) for some t\in[0,x].

    f'(t)=\frac{1}{2\sqrt{1+x}}. For x>0, f'(t)<\frac{1}{2}.

    Thus,

    \frac{\sqrt{1+x}-1}{x}<\frac{1}{2}\implies\sqrt{1+x}<1+\frac{x}{2} for x>0

    as desired.

    --------

    Using the same method, see if you can generalize that result.
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