# Prove uniqueness of solution to a simple equation

1. Jan 12, 2009

### Irid

1. The problem statement, all variables and given/known data

Prove that the equation $$e^x = 1+x$$ admits the unique solution $$x_0 = 0$$.

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

$$x>-1$$

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. Jan 12, 2009

### Dick

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.