Question

Show that when the mean value theorem is applied to any interval [a, b], the point c which satisfies the conclusion of the theorem is the midpoint of the interval

Answer 1

The proposition stated here cannot be shown in general, because it is not true in general.

Consider the attached, which shows that over the interval [0, 4] for function
  f(x) = x(x-4)²
the value of c that satisfies the theorem is c=4/3, which is not the midpoint of the interval.