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Prove uniqueness of solution to a simple equation

  1. Jan 12, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove that the equation [tex]e^x = 1+x[/tex] admits the unique solution [tex]x_0 = 0[/tex].

    2. The attempt at a solution

    I think there should be a very simple proof based on monotonicity or the absence of inflection points, etc.

    But I have no idea how to do it, and what theorems are to be used. All I can say from the equation, is that if there are solutions, they certainly satisfy

    [tex] x>-1[/tex]

    I'm actually a little ashamed that I can't do this, most likely, trivial problem, maybe somebody can show me the right path?
  2. jcsd
  3. Jan 12, 2009 #2


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    Homework Helper

    1+x is the tangent line to e^x at x=0 and e^x is concave up. Or note that if 1+x intersected e^x at another point then the Mean Value Theorem would say there is a point in between where the derivative of e^x is 1.
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