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
If you may use that $\displaystyle f(x) = x^2e^{|x|}-2$ is continuous, it shouldn't be that hard: $\displaystyle f[a,b]\to \mathbb{R}$ must reach all values in $\displaystyle [f(a), f(b)]$.
(look up theorem of Bolzano. I believe there's other even other names for that)
given that $\displaystyle f(1) = e-2 > 0$ and $\displaystyle f(1/2) < 0$ it follows that there exists a $\displaystyle x_0$ with $\displaystyle 1/2 < x_0 < 1$ such that $\displaystyle f(x_0) = 0$.
You can use this argument twice to show the existence of 2 roots (without necessarily finding them).