In a bounded function, there must be at least one instantaneous slope that is equal to average slope from the upper bound to the lower bound.
For example, on a function f(x)= x^2, bounded from x=-1 to x=1, the average slope is 0. The instantaneous rate of change at x=0 is also 0. Thus, the example f(x) = x^2 supports the mean value theorem.
For example, on a function f(x)= x^2, bounded from x=-1 to x=1, the average slope is 0. The instantaneous rate of change at x=0 is also 0. Thus, the example f(x) = x^2 supports the mean value theorem.