The change of sign rule is a rather simple idea. If we are given a continuous function $f(x)$ and we know that, for some numbers $a<b,$

then we must have $f(x)=0$ at some point $a<x<b.$

Looking at a picture makes this clear:

If $f(a)$ is negative and $f(b)$ is positive, there must be a root somewhere in-between. (This, of course, also works if $f(a)>0$ and $f(b)<0.$

The condition that $f$ is continuous is important, as you will see in the problems. For now, though, let's get used to using the change of sign rule.

1. Show that, for some $x$ where $0.8\le x\le 1.0,$ the line

   $$y=3x-2$$

   and the curve

   $$y=\cos \left(x\right)$$

   intersect.

2. By using a suitable interval, determine the $x$ coordinate of the point of intersection correct to $2$ significant figures.