I made this question here Show $a e^x=1+x+\frac{x^2}{2}$ has exactly one real root but I wrote the equation wrong. I reproduce the post with the correct equation:
I'm struggling with the following problem:
Show that the equation $a e^x=1+x+\frac{x^2}2$, where $a$ is a positive constant, has exactly one real root.
Looking at the graphs of both $ae^x$ and $1+x+\frac{x^2}2$, they will intersect on the left half of the plane when $a>1$ and on the right half when $a<1$ (because a makes $e^x$ grow faster or slower). So if I could find a point $x_1$ where $a e^{x_1}-(1+x_1+\frac{{x_1}^{2}}2)<0$ and a point $x_2$ where $a e^{x_2}-(1+x_2+\frac{{x_2}^2}2)>0$, by the intermediate value theorem, the equation would have a real root. Also,since the derivative of $a e^x-(1+x+\frac{x^2}2)$ is always positive, I could colclude that there can't exist another real root.
So my question is, how can I find $x_1$ and $x_2$? I can't figure out how to express $x$ in terms of $a$.