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For the sake of argument, suppose that `f(a)` is smaller
than `f(b)`. Then `f` can't increase from
`f(a)` to `f(b)` unless at some points it has slope *at least*
as great as the slope of the straight line joining the point
`(a,f(a))` to the point `(b,f(b))`.

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?