Question

The goal of this problem is to show that the function f(x) = 3x³ - 6x satisfies both of the conditions (the hypotheses) and the conclusion of the Mean Value Theorem for x in the interval [-3, 3]. Verification of Hypotheses: Fill in the blanks to show that the hypotheses of the Mean Value Theorem are satisfied: f(x) is continuous on [-3, 3] and is differentiable on (-3, 3). Note: The answer in each box should be one word. Verification of the Conclusion: If the hypotheses of the Mean Value Theorem are satisfied, then there is at least one c in the interval (-3, 3) for which f'(c) = (f(3) - f(-3)) / (3 - (-3)). Verify that the conclusion of the Mean Value Theorem holds by computing (f(3) - f(-3)) / (3 - (-3)) = Now find c in (-3, 3) so that f'(c) equals the answer you just found. c1, c2 = Note: For this problem there are two correct values of c, we call them c1, c2. Enter them separated by a comma. Note: On an exam you may be asked to state the Mean Value Theorem (i.e., it may not be given to you), and to verify that a given function satisfies the assumptions of the Mean Value Theorem.

          The goal of this problem is to show that the function

f(x) = 3x³ - 6x

satisfies both of the conditions (the hypotheses) and the conclusion of the Mean Value Theorem for x in the interval [-3, 3].

Verification of Hypotheses: Fill in the blanks to show that the hypotheses of the Mean Value Theorem are satisfied:

f(x) is continuous on [-3, 3] and is differentiable on (-3, 3).

Note: The answer in each box should be one word.

Verification of the Conclusion: If the hypotheses of the Mean Value Theorem are satisfied, then there is at least one c in the interval (-3, 3) for which

f'(c) = (f(3) - f(-3)) / (3 - (-3)).

Verify that the conclusion of the Mean Value Theorem holds by computing

(f(3) - f(-3)) / (3 - (-3)) = 

Now find c in (-3, 3) so that f'(c) equals the answer you just found.

c1, c2 = 

Note: For this problem there are two correct values of c, we call them c1, c2. Enter them separated by a comma.

Note: On an exam you may be asked to state the Mean Value Theorem (i.e., it may not be given to you), and to verify that a given function satisfies the assumptions of the Mean Value Theorem.

The goal of this problem is to show that the function

f(x) = 3x³ - 6x

satisfies both of the conditions (the hypotheses) and the conclusion of the Mean Value Theorem for x in the interval [-3, 3].

Verification of Hypotheses: Fill in the blanks to show that the hypotheses of the Mean Value Theorem are satisfied:

f(x) is continuous on [-3, 3] and is differentiable on (-3, 3).

Note: The answer in each box should be one word.

Verification of the Conclusion: If the hypotheses of the Mean Value Theorem are satisfied, then there is at least one c in the interval (-3, 3) for which

f'(c) = (f(3) - f(-3)) / (3 - (-3)).

Verify that the conclusion of the Mean Value Theorem holds by computing

(f(3) - f(-3)) / (3 - (-3)) =

Now find c in (-3, 3) so that f'(c) equals the answer you just found.

c1, c2 =

Note: For this problem there are two correct values of c, we call them c1, c2. Enter them separated by a comma.

Note: On an exam you may be asked to state the Mean Value Theorem (i.e., it may not be given to you), and to verify that a given function satisfies the assumptions of the Mean Value Theorem.

Added by Jhoanna H.

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