The Mean Value Theorem : If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that 𝑓′(𝑐) = 𝑓(𝑏) − 𝑓(𝑎)
.
𝑏 − 𝑎
If the conditions of the theorem are met, then there is a point on the graph of the function with tangent line parallel to the line passing through the endpoints of the graph of the function. Note:
Rolle’s Theorem is actually a special case of the Mean Value Theorem.
Example: page 217, exercise 42. Let 𝑓(𝑥) = −𝑥 2 − 𝑥 + 6, 𝑓𝑜𝑟 2 ≤ 𝑥 ≤ 2.
Are the conditions of the MVT satisfied? The function is a polynomial defined on a closed interval, [-2,2]. A polynomial function is continuous on any closed interval (p. 95).
Polynomial functions are differentiable at any point (Theorems 3.2, 3.3, 3.4, 3.5). Thus f is differentiable on (-2,2), and 𝑓 ′ (𝑥) = −2𝑥 − 1, 𝑓𝑜𝑟 − 2 < 𝑥 < 2.
Thus, the conditions of the theorem are satisfied. Therefore, there is a number c between -2 and
2 such that the line tangent to the graph of the function at the point (c, f(c)) is parallel to the line passing through the two endpoints of the graph.
At the left-hand endpoint of the interval, we have 𝑓(−2) = −(−2) 2 − (−2) + 6 = 4.
At the right-hand endpoint of the interval, we have 𝑓(2) = −(2) 2 − (2) + 6 = 0.
The line passing through the points (-2, 4) and (2, 0) has slope
0 − 4 𝑚 =
2 − (−2)
= −1, and equation 𝑦 − 0 = −1(𝑥 − 2), or

𝑦 = −𝑥 + 2.
By the MVT, there must be some number c between -2 and 2 such that the line passing through the point (c, f(c)) has slope -1.
From the equation for the derivative, we have 𝑓 ′ (𝑐) = −2𝑐 − 1 = −1, which implies that c = 0. At c = 0, we have f(0) = 6. The line passing through the point (0, 6) and having slope -1 must be tangent to the graph of f at the point (0, 6). The equation for this line is 𝑦 − 0 = −1(𝑥 − 6), or 𝑦 = −𝑥 + 6.
To verify that the line tangent to the graph of f at c = 0 is parallel to the line through the two endpoints of the graph, let us graph the function and the two lines together, as shown below.
The lower line is the line passing through the two endpoints of the graph of the function. The upper line is the line tangent to the graph of the function at the point (0, 6).