MTH132
2
Chapter 3 - Applications of Differentiation
Michigan State University
The Mean Value Theorem
Theorem 2.3 (Mean Value Theorem). Let f (x) be a function which satisfies the following two properties:
Theorem 2.1 (Rolle’s Theorem). Let f (x) be a function
which satisfies the following three properties:
(1) f (x) is continuous on the interval [a, b]
(1) f (x) is continuous on the interval [a, b]
(2) f (x) is differentiable on (a, b)
(2) f (x) is differentiable on (a, b)
Then there is a number c in (a, b) such that
(3) f (a) = f (b)
f 0 (c) =
Then there is a number c in (a, b) such that f 0 (c) = 0.
Remark 2.2. The conclusion of Rolle’s Theorem says
that if the function values agree at the endpoints, then
there is a place in between where the tangent line is horizontal.
f (b) − f (a)
b−a
Remark 2.4. The conclusion of The Mean Value
Theorem says that there is a place in the interval where
the tangent line is parallel to the secant line between the
endpoints.
Remark 2.5. Notice that Rolle’s Theorem and The Mean Value Theorem tell you that “there exists” a number c
with certain properties, but neither theorem tells you what that the value of c is, or how to find it.
Picture:
Corollary 2.6. If f 0 (x) = 0 for all x in an interval (a, b), then f (x) must be constant on (a, b).
Corollary 2.7. If f 0 (x) = g 0 (x) for all x in an interval (a, b), then f (x) = g(x) + c for some constant c.
3
(a) - Derivatives and Graphs
Theorem 3.1.
(a) If f 0 (x) > 0 on (a, b), then f (x) is increasing on (a, b).
(b) If f 0 (x) < 0 on (a, b), then f (x) is decreasing on (a, b).
Theorem 3.2 (First Derivative Test). Suppose that f (x) is a function and that c is a critical point (or “critical number”)
of f (x).
(a) If f 0 (x) changes from positive to negative at x = c, then f (x) has a local maximum at x = c.
(b) If f 0 (x) changes from negative to positive at x = c, then f (x) has a local minimum at x = c.
(c) If f 0 (x) does not change sign at x = c, then f (x) has neither a local maximum nor a local minimum at x = c.
1
MTH132
Chapter 3 - Applications of Differentiation
Michigan State University
Example 3.3 (Instructor).
For each of the following functions (on the given domain), tell whether the hypotheses of the Mean Value Theorem are
satisfied. If so, try to find the value of c guaranteed by the theorem.
(a) f (x) = |x| on the domain [−3, 3]
(b) f (x) = x2 + 3x + 5 on the domain [0, 1]
Example 3.4 (Instructor).
Suppose that f (x) is continuous and differentiable everywhere, with f (0) = 5, and f 0 (x) ≤ 2 for all x. What is the largest
that f (3) could be?
Example 3.5 (Instructor).
Prove that the equation x3 + x + 1 = 0 has exactly one solution.
Example 3.6 (Instructor).
For the following functions, find the intervals on which it is increasing and decreasing, and find where the local maximum and
local minimum values occur.
(a) f (x) = 2x3 + 3x2 − 36x on the domain (−∞, ∞)
(b) f (x) = cos2 (x) − 2 sin(x) on the domain [0, 2π]
Example 3.7 (Student).
For each of the following functions (on the given domain), tell whether the hypotheses of the Mean Value Theorem are
satisfied. If so, try to find the value of c guaranteed by the theorem.
(a) f (x) =
1
x
on the domain [1, 3]
(b) f (x) = x3 − 3x + 2 on the domain [−2, 2]
Example 3.8 (Student).
Suppose that f (x) is continuous and differentiable everywhere, with f (8) = 30, and f 0 (x) ≤ 1 for all x. What is the largest
that f (10) could be?
Example 3.9 (Student).
Prove that the equation 2x + cos(x) = 0 has exactly one solution.
Example 3.10 (Student).
For the following functions, find the intervals on which it is increasing and decreasing, and find where the local maximum and
local minimum values occur.
(a) f (x) =
x
x2 +1
on the domain (−∞, ∞)
(b) f (x) = sin(x) + cos(x) on the domain [0, 2π]
2