The goal Of this problem is to show that the function f(x) =18 3x3 4x+4 satlsfles both the conditlons (the hypotheses) and the conclusion of the Mean Value Theorem for x} in the interval [3, ]. Verification 01' Hypotheses: Fill in the blanks to show that the hypotheses ot the Mean Value Theorem are satisfied: x) is continuous E on [3. OJ and is differentiable I on (-3, 0). Note: The answer in each box should be one word Verification 01' the Conclusion: If the hypotheses of the Mean Value Theorem are satised. then there is at least one c in the interval (3, {J} for which , _ M f k) 07(73) - Verify that the conclusion Of the Mean value Theorem holds by computing flit) - f(3) _ 0 (3) 7 2 Now nd c in (3, 0) so that f'(c) equals the answer you just found. For this problem there is only one correct value of c. c = 2 Note: On an exam you may be asked to state the Mean Value Theorem tie, it may not be given to you}, and to verify that a given function satises the assumptions of the Mean Value Theorem. The goal of this problem is to show that the function f(x) = 3 -8x2 satisfies both the conditions (the hypotheses) and the conclusion of the Mean Value Theorem for x) in the interval [-4, 3]. Verification of Hypotheses: Fill in the blanks to show that the hypotheses of the Mean Value Theorem are satisfied: f(x) is continuous E on [-4, 3] and is differentiable E on (-4, 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 (-4, 3) for which f'(c) = 1(3) - f(-4) 3 - (-4) Verify that the conclusion of the Mean Value Theorem holds by computing f(3) - f(-4) -130/7 E 3 - (-4) Now find c in (-4, 3) so that f' (c) equals the answer you just found. (For this problem there is only one correct value of c.) E 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 it satisfies both 01 the conditions (the hypotheses) and the conclusion of the Mean Value Theorem for x in the interval [2, 2]. Verification 01' Hypotheses: Fill in the blanks to show that the hypotheses of the Mean Value Theorem are satisfied: x) is I on [12] and is I on(2,2). Note: The answer in each box should be one word. Verification 01' the Conclusion: It the hypotheses ol' the Mean Value Theorem are satised, then there is at least one c in the interval (2, 2) for which , _ w f (0 2_(_2) . Verify that the conclusion Of the Mean Value Theorem hOIGS by computing n2) f(2) = z 2 (2) Now nd c in (2, 2) so that f'(c) equals the answer you Just found. CI. 61 = 2 Note: For this problem there are two correct values of c, we call them cl, c2. Enter them separated by a comma. Note: On an exam you may be asked to state the Mean Value Theorem li.e., it may not be given to you), and to verify that a given function satises the assumptions of the Mean Value Theorem. The goal 01' this problem is to snow that the function f()i:)=)i:+2 x 1 satisfies both of the conditions (the hypotheses) and the conclusion of the Mean Value Theorem for x in the interval [ . 2] i Venflcation o1 Hypotheses: Fill in the blanks to show that the hypotheses Of the Mean Value Theorem are satisfies: f(x)is I on [all andis 2 on ($2). NON! The answer lh each box should be one word. 1 Verification M the COHCILISIOI'I: It the hypotheses of the Mean Value Theorem are satised, then there is at least one C In the interval (a, 2) her which 1 re) f (5) I'M = - 2 , 1 2 Verify that the conclusion of the Mean Value Theorem holds by computing fa) f (i) 2 = 2 2 l 2 1 Now nd 0 in (E , 2) so that f'(c) equals the answer you just found. (For this problem there is only one correct value of c.) c: 2 Note: On an exam you may be asked to state the Mean Value Theorem (ie., it may not be given to you}, and to verify that a given function satises the assumptions of the Mean Value Theorem. Consider the function f(x) = x - 8x + 4 on the interval [0, 8]. Verify that this function satisfies the three hypotheses of Rolle's Theorem on the inverval. Fill in the first two blanks below with the appropriate words. f(x) is E on [0, 8] f(x) is on (0, 8) and f(0) = f(8) = I Then by Rolle's theorem, there exists a c such that f (c) = 0. Find the value c. CEThe goal of this problem is to show that the function f(x) : \\f? x satisfies both of the conditions {the hypotheses) and the conclusion of the Mean Value Theorem for x in the interval [7, '4']. Verification of Hypotheses: Fill in the blanks to show that the hypotheses of the Mean Value Theorem are satisfies: x) is I on [17] and is 2 on (17). Note: The answer In each box should be one word. Verification of the Conclusion: It the hypotheses of the Mean Value Theorem are satised. then there is at least one( , c] in the interval (7, 7} for which ?) - f(-7) f (c): 7-(7) Verify that the conclusion Of the Mean Value Theorem holes by computing rm 7 fET) = 7 (T) I Now nd 6 in (i', 7) so that f'(c) equals the answer you just tound. (For this problem there is only one correct value of c.) c = 2 Note: On an exam you may be asked t0 state the Mean Value Theorem (i.e., it may not be given to you), and t0 verify that a given function satises the assumptions of the Mean Value Theorem