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Math Help - Proving an inequality using the mean value theorem

  1. #1
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    Proving an inequality using the mean value theorem

    The question:
    By using the mean value theorem, show that 1 + x < e^x whenever x > 0.

    My attempt:
    I took e^x as the function, and [0,x] as the interval.

    \frac{f(x) - f(0)}{x} = e^c

    \frac{e^x - 1}{x} = e^c

    e^c > 1 (because c must be in the interval (0, x), and e^0 = 1)

    \frac{e^x - 1}{x} > 1
    e^x - 1 > x //we can do this since we know x is positive
    e^x > x + 1
    Therefore x + 1 < e^x

    Is this correct? Thanks!
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  2. #2
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    Opalg's Avatar
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    Completely correct!
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  3. #3
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    Awesome! So I'm not completely hopeless after all. :P

    Thanks.
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