Answer
1 number
explanation
if we use a graphing calculator, there is only one such number
that satisfies the conclusion of the mean value theorem, that is, a number
such that
![f'(c) = \dfrac{f(0.75) - f(0.5)}{0.75-0.5},](/tex.php?f=f'(c) = \dfrac{f(0.75) - f(0.5)}{0.75-0.5},)
where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.
in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.