# Thread: Show that x^2*e^|x|=2 has two solutions

1. ## Show that x^2*e^|x|=2 has two solutions

How would I do this? I know from the graph that the roots are -0.9 and 0.9 but how do I show this??? Any help much appreciated.
Thank you

2. If you may use that $f(x) = x^2e^{|x|}-2$ is continuous, it shouldn't be that hard: $f[a,b]\to \mathbb{R}$ must reach all values in $[f(a), f(b)]$.

(look up theorem of Bolzano. I believe there's other even other names for that)

given that $f(1) = e-2 > 0$ and $f(1/2) < 0$ it follows that there exists a $x_0$ with $1/2 < x_0 < 1$ such that $f(x_0) = 0$.

You can use this argument twice to show the existence of 2 roots (without necessarily finding them).

3. I meant the Intermediate value Theorem. It says: If f is contunious on a closed interval $f:[a,b]\to \mathbb{R}$ then for any $c\in [f(a),f(b)]$ exists a $x_0\in [a,b]$ such that $f(x_0)=c$.