Math 2413
Notes 3.2
Section 3.2 – The Mean-Value Theorem
Definition: If there is an open interval I containing c in which either f (c) ≤ f(x) for all x in I , or f (c) ≥ f(x) for
all x in I , then we say that f(c) is a local extreme value of f .
In the former case, f has a local minimum at x = c, and in the latter case it has a local maximum at x = c.
Theorem 4.2.1: If f (c) is a local extreme value and if f is differentiable at x = c, then f ' (c) = 0.
Theorem 4.2.2: Rolle’s Theorem
Suppose that f is continuous on the closed interval [a,b] and differentiable on the open interval (a, b) .
If f (a) = f (b), then there is at least one number c in (a,b) for which f ' (c) = 0.
Rolle’s theorem is sometimes stated as follows: Suppose that f is continuous on the closed interval [a,b] and
differentiable on the open interval (a,b). If f (a) = f (b) = 0, then there is at least one number c in (a,b) for
which f ' (c) = 0 .
That is, Rolle’s theorem tells us that between any two roots of f , there must be a root of f ' .
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Math 2413
Notes 3.2
3
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Example 1: Use Rolle’s theorem to prove that the equation x  3 x  10 x  12  0 has exactly one real solution
over the interval [0,2].
Theorem 4.2.3: The Mean-Value Theorem
Suppose that f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b). There is at
least one number c in (a,b) for which f ' (c ) 
f (b)  f (a )
.
ba
Example 2: Verify that the function f ( x)  3 x 2  2 x  5 satisfies the conditions of the mean-value theorem over
the interval [-1, 1] , and find all numbers c that satisfy the conclusion of the theorem.
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Math 2413
Example 3: Verify that the function f ( x) 
Notes 3.2
x
satisfies the conditions of the mean-value theorem over the
x2
interval [1, 4] , and find all numbers c that satisfy the conclusion of the theorem.
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