How can I find the Estimate the value(s) of c that satisfy the conclusion of the Mean Value Theorem on the interval [2, 6].

Question

The question presents only the graph and since I don't know the equation, I don't know how to estimate it.

1. information of the graph

g is differentiable on the open interval (0, 8)

g is continuous on the closed interval [0, 8]

f(0)= 1, f(8)= 3, f(2)= 3, f(6)= 1

Q. Estimate the value(s) of c that satisfy the conclusion of the Mean Value Theorem on the interval [2, 6]. (Enter your answers as a comma-separated list. Round your answers to one decimal places. If an answer does not exist, enter DNE.)

I know the slope between 2 and 6 is -1/2. How can I find the estimated value of c?

Answer 1

I believe the key is in the direction to "estimate" to value of "c". Draw a line (with slope -1/2) that goes between (2, 3) and (6, 1). Draw a series of parallel lines and look for one (or more) that is/are tangent to the function curve. Draw the tangent point(s). The approximate x-value of the tangent point(s) is/are the value(s) of "c".

This example shows three values of "c":