Mean Value Theorem and a parabola:

Apply the Theorem to the points at x = 2 and x = 5 on the parabola

We first need to find the slope of the secant line through (2, f(2) ) and (5, f(5) ):

To find the intermediate point (c, f(c) ), we need f '(x):

The point at c is defined as having a tangent line with the same slope, 1, as the secant line.
f '(x) gives the slope at any x. We can evaluate it at the (unknown) point at x = c:

. Hence c = 7/2, which indeed is between the other two points.

Test Problem:

Apply the Mean Value Theorem to this function and the points at x = 1 and x = 5.

a) 3
b) 5
c) 3.5
d) 4.5