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Math Help - Lagrange's Mean Value Theorem

  1. #1
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    Lagrange's Mean Value Theorem

    Can u plzz solve this question... Using Lagrange mean value theorem show that e Rest to x > 1 + x, x > 0.
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  2. #2
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    I think you wrote
    e^x>1+x, x>0
    Assume what you are trying to prove is not true ie. there exists y>0 such that e^y<1+y. Then the mean value theorem tells us that for some  x = c\in(0,y), \frac{d}{dx}e^x<\frac{1+y-1}{y}. Find a contradiction.
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  3. #3
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    Please is there any other way

    Please sir is there any other weay because this is very confussing .If possible can u give me a reference from where i can get a complete explaination about hw u came to this conclusion....
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  4. #4
    Senior Member bkarpuz's Avatar
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    Exclamation

    Quote Originally Posted by Shivanand View Post
    Can u plzz solve this question... Using Lagrange mean value theorem show that e Rest to x > 1 + x, x > 0.
    Clearly, \mathrm{e}^{x} is positive, increasing and convex for x\geq0.
    Then, for x\geq0, the function \mathrm{e}^{x} is always above its tangent lines; i.e., \mathrm{e}^{x}\geq \mathrm{e}^{x_{0}}(x-x_{0})+\mathrm{e}^{x_{0}} for any x,x_{0}\geq0 (recall that a tangent line of a function f at the point x_{0} is given by g(x):=f^{\prime}(x_{0})(x-x_{0})+f(x_{0})).
    Letting x_{0}=0, we get the desired result.
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