A student's working is shown below.

Consider the curve

$$y=\frac{199e^{x^2}-200}{200e^{x^2}}$$

Let $y=f(x)$. We have

$$\begin{array}{rl}f(-0.1)& =0.0049502\\ f(0.1)& =0.0049502\end{array}$$

Since both values are positive, the curve has no roots between $-0.1$ and $0.1$.

1. Give a simple counter-example to demonstrate that the student's reasoning is not generally true.

2. By considering the value of $f$ on some other interval, prove that $f$ does, in fact, have a root within $-0.1<x<0.1.$

For (a), something like $y=x^2-4$ would work, so long as you explain why this is a counter-example.

For (b), try $f(0).$

Give the root of $f$ correct to $2$ decimal places.