root of $ce^x=1+x+\frac{x^2}{2}$, $c>0$

Question

Given that $c>0$, How can I Prove that $ce^x=1+x+\frac{x^2}{2}$ has exactly only one root?

Answer 1

Hint:   write the equation as $2c = f(x) = \dfrac{x^2+2x+2}{e^x}\,$, then show that:

• $\,f\,$ is strictly decreasing on $\mathbb{R}\,$

• $\,f(\mathbb{R})=\mathbb{R}^+\,$

It follows that $\,f : \mathbb{R} \to \mathbb{R}^+\,$ is a bijection, so the equation $\,2c=f(x)\,$ has a unique solution $\,x = f^{-1}(2c)\,$ for every $\,c \in \mathbb{R}^+\,$.