The goal of this problem is to show that the function f(x) =5 -3x2 satisfies both the conditions (the hypotheses) and the conclusion of the Mean Value Theorem for x) in the interval [-1, 3]. Verification of Hypotheses: Fill in the blanks to show that the hypotheses of the Mean Value Theorem are satisfied: f(x) is on [-1, 3] and is E on (-1, 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 (-1, 3) for which f' ( c ) = f(3) - f(-1) 3 - (-1) Verify that the conclusion of the Mean Value Theorem holds by computing f(3) - f(-1) 3 - (-1) Now find c in (-1, 3) so that f' (c) equals the answer you just found. (For this problem there is only one correct value of c.) C =The goal of this problem is to show that the function f(x) = x' - 3x + 4x + 1 satisfies both the conditions (the hypotheses) and the conclusion of the Mean Value Theorem for x) in the interval [-2, 6]. Verification of Hypotheses: Fill in the blanks to show that the hypotheses of the Mean Value Theorem are satisfied: f(x) is Z on [-2, 6] and is on (-2, 6). 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 (-2, 6) for which f'(c) = f(6) - f(-2) 6 - (-2) Verify that the conclusion of the Mean Value Theorem holds by computing f(6) - f(-2) E 6 - (-2) Now find c in (-2, 6) so that f' (c) equals the answer you just found. For this problem there is only one correct value of c. C= EThe goal of this problem is to show that the function f(x) = 5x3- 2x satisfies both of the conditions (the hypotheses) and the conclusion of the Mean Value Theorem for a in the interval [-4, 4]. Verification of Hypotheses: Fill in the blanks to show that the hypotheses of the Mean Value Theorem are satisfied: f(x) is on [-4, 4) and is E on (-4, 4). 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, 4) for which f' (c) = f(4) - f(-4) 4 -(-4) Verify that the conclusion of the Mean Value Theorem holds by computing f(4) - f( -4) 4- (-4) Now find c in (-4, 4) so that f' (c) equals the answer you just found. (1 , C2 = EThe goal of this problem is to show that the function f(x) = =+ 3 satisfies both of the conditions (the hypotheses) and the conclusion of the Mean Value Theorem for x in the interval 00 1 1 Verification of Hypotheses: Fill in the blanks to show that the hypotheses of the Mean Value Theorem are satisfied: f(x) is E on 2,8 and is E on 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 $18 for which f(8) - f 00 | H f' (c) = 8 Verify that the conclusion of the Mean Value Theorem holds by computing f(8) - 8 Now find c in 8 ' 8 so that f' (c) equals the answer you just found. (For this problem there is only one correct value of c.)Consider the function f(x) = x2 - 4x + 4 on the interval [0, 4]. 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 on [0, 4] f(x) is on (0, 4) and f(0) = f(4) = Then by Rolle's theorem, there exists a c such that f (c) = 0. Find the value c. C= EThe goal of this problem is to show that the function f(x) = V3 -1 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(r) is on [-3, 3] and is 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) E 3- (-3) Now find c in (-3, 3) so that f' (c) equals the answer you just found. (For this problem there is only one correct value of c.) C :The goal of this problem is to show that the function f(x) = 5x - 4x satisfies both of the conditions (the hypotheses) and the conclusion of the Mean Value Theorem for + in the interval |9, 16]- Verification of Hypotheses: Fill in the blanks to show that the hypotheses of the Mean Value Theorem are satisfied: f(x) is on 9, 16] and is on (9, 16). 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 (9, 16) for which f' (c) = f(16) - f(9) 16 - (9) Verify that the conclusion of the Mean Value Theorem holds by computing f(16) - f(9) E 16 - (9) Now find c in (9, 16) so that f'(c) equals the answer you just found. (For this problem there is only one correct value of c.) C