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In this case, the function attains its maximum value at one
of the endpoints of the interval on which it is defined.
Lemma 6.13
needs the function to be differentiable **as a function on** `D` at the
point at which it attains its maximum. This means that `f` has to attain its
maximum at an interior point of `D`. (Remember that part of the
definition of differentiability
is that the point in question has to be an interior point of the domain.)

Thus, although `f` is differentiable, as a function on the whole real
line, at the point `x=2` it is not differentiable as a function on
`[1,2]`. Yet we can only call `x=2` a maximum of `f` as a function on `[1,2]`, which explains why it is inappropriate to apply Lemma 6.13
to this situation.

In fact, the purpose, from a technical point of view, of the requirement that
`f(a)=f(b)` in Rolle's Theorem
is precisely in order to ensure that at least one out of the maximum or minimum
of the function lies in the interior of the interval, so that we *can* apply
Lemma 6.13.