IVT, MVT, and Rolle's Theorem Question 5

The function $f$ is continuous for all real numbers, and the average rate of change of $f$ on the closed interval $[2,5]$ is $-\dfrac{5}{2}$ . For $2\lt c\lt 5$ , there is no value of $c$ such that $f'(c)=-\dfrac{5}{2}$ . Of the following, which must be true?

\frac{1}{3}\int\limits_2^5 f(x)  dx = -\frac{5}{2}

\int\limits_2^5 f(x)   dx

does not exist

\dfrac{f'(2)+f'(5)}{2}=-\dfrac{5}{2}

f

is not differentiable on the open interval

(2,5)

Approach

The mean value theorem gurantees that there must be a point where the instantaneous rate of change is equal to the average rate of change. For this to be true however, the function needs to be differentiable on the interval.

The definite integrals exist as long as the function is continuous, but it does not necessarily have to equal $-\frac{5}{2}$ .