Question: Draw a picture to show a possible curve for f, together with this straight line
Question: What is the slope of the straight line?
Likewise, if f is to increase from f(a) to f(b) over the interval, then its slope can't always be greater than the slope of the straight line joining (a,f(a)) to the point (b,f(b)).
It's tempting to use this argument to conclude at once that since f' has to be bigger that than the slope of the straight line joining (a,f(a)) to (b,f(b)) at some points, and less than it at others, it must be exactly equal to this slope at some point.
Question: What's the gap in this argument?
Question: What theorem might you want to apply to fill that gap?
Question: Why can't you?