Answer Key Testname: 1325 TEST 3 REVIEW 2) f(x) =2x4 - 8x3 Find the Critical Numbers and use them to find the endpoints of the Test Intervals. f'(x) = 8x3 - 24x2 = Set f'(x) = 0 = 8x3 - 24x2 = 0 = 8x2(x - 3) = 0 = 8x2 = 0, x - 3 =0 = x = 0, x = 3 (CNs) Use this table to help with the First Derivative Test: Test Intervals (-00, 0) (0, 3) (3, 00) Test Points (x) -1 1 4 f' (x) = 8x3 - 24x2 f'(-1) = -32 f'(1) = -16 f'(4) = 128 Sign of f' (x) Incr 71 / Decry yyy Local Extrema min (3, f(3)) = (3, -54) Increasing Intervals: (3, 0.) Decreasing Intervals: (-00, 0), (0, 3) Local Minima: (3, -54) Local Maxima: None 3) f"(x) = 14 4) f"(x) = 18x - 8 5) f"(x) = 72x2 - 14 MATH 1325 - Test 3 Review 8 Dave Rice7) f(x) =x4 + 4x3 Find the Possible Inflection Points and use them to find the endpoints of the Test Intervals. Use this table to help with the Concavity Test: Test Intervals Test Points (x) f" (x) Sign of f" (x) CC Up U / CC Down n Inflection Points CC Up Intervals: CC Down Intervals:. Inflection Points: Objective: (5.3) Determine Concavity from Equation MATH 1325 - Test 3 Review 4 Dave RiceAnswer Key Testname: 1325 TEST 3 REVIEW 6) f(x) =-2x3 - 18x2 - 4x + 7 Find the Possible Inflection Points and use them to find the endpoints of the Test Intervals. f'(x) = -6x2 - 36x - 4 = f"(x) =-12x - 36 = Set f"(x) =0 = -12x - 36 =0 = -12x =36 = x= -3 (Possible Inflection Points) Use this table to help with the Concavity Test: Test Intervals (-00, -3) (-3, 00) Test Points (x) -4 0 f" (x) = -12x - 36 f"(-4) = 12 f"(0) = -36 Sign of f" (x) + CC Up U / CC Down n U n Inflection Points (-3, f(-3)) = (-3, -89) CC Up Intervals: (-00, -3) CC Down Intervals: (-3, co) Inflection Points: (-3, -89) MATH 1325 - Test 3 Review 9 Dave RiceUse the Second Derivative Test to find the Relative (Local) Maxima and Minima of f(x). 8) f(x) =3x4 - 8x3 - 18x2 + 10 f' (x) = Find the Critical Numbers (CNs): CNs: f"(x) = Plug the CNs into f"(x) and perform the Second Derivative Test: Relavtive (Local) Minima in (x,y) format:. Relavtive (Local) Maxima in (x,y) format: Objective: (5.3) Determine If Critical Number is Maximum, Minimum, or Neither MATH 1325 - Test 3 Review 5 Dave RiceThe profit Pbd in dollars from the daily sale of x laptops is given by: P{X} = x3 + 3x2 + 72x + 100. Use the Second Deriviative Test to nd the total number of laptops that should be sold in one day in order to maximize the profit for that day. Also, nd the maximum profit. 9} f(x) = x3 + 3x2 + 72:: + 100 FIX) = Find the Critical Numbers (CNS): CNS: f"(xl = Plug the CNS into f"(x] and perform the Second Derivative Test: units must be sold each day to have a maximimum profit of :5 Objective: (5.3] Determine 1f Critical Number 15 Maximum, Minimum, er Neither MATH 1325 Test 3 Review 6 Dave Rice 2 ) f ( x) = 2x4 - 8x3 Find the Critical Numbers and use them to find the endpoints of the Test Intervals. Use this table to help with the First Derivative Test: Test Intervals Test Points (x) f' (x) Sign of f' (x) Incr 71 / Decry Local Extrema Increasing Intervals: Decreasing Intervals: Local Minima: Local Maxima: Objective: (5.2) Find Location and Values of Extrema from Equation MATH 1325 - Test 3 Review N Dave RiceAnswer Key Testname: 1325 TEST 3 REVIEW 7) f ( x) = x4+4x3 Find the Possible Inflection Points and use them to find the endpoints of the Test Intervals. f'(x) = 4x3 + 12x2 = f"(x) =12x2 + 24x = Set f"(x) = 0 = 12x2 + 24x = 0 = 12x(x + 2) = 0 = 12x = 0, x + 2 =0 = x = 0, x = -2 (Possible Inflection Points) Use this table to help with the Concavity Test: Test Intervals (-00, -2) (-2, 0) (0, 00) Test Points (x) -3 -1 1 f" (x) = 12x2 + 24x f"(-3) = 36 f"(-1) = -12 f"(1) = 36 Sign of f" (x) + CC Up U / CC Down n U n U Inflection Points (-2, f(-2)) = (-2, -16) (0, f(0)) = (0, 0) CC Up Intervals: (-00, -2), (0, co) CC Down Intervals: (-2, 0) Inflection Points: (-2, -16), (0, 0) MATH 1325 - Test 3 Review 10 Dave RiceUse the First Derivative Test to find the Relative (Local) Maxima and Minima of f(x). 1) f(x) = 3x4 - 4x3 - 36x2 + 12 Find the Critical Numbers and use them to find the endpoints of the Test Intervals. Use this table to help with the First Derivative Test: Test Intervals Test Points (x) f' (x) Sign of f' (x) Incr 71 / Decry Local Extrema Increasing Intervals: Decreasing Intervals: Local Minima:. Local Maxima: Objective: (5.2) Find Location and Values of Extrema from Equation MATH 1325 - Test 3 Review Dave RiceFind the second deriviative f"(x) for the function f(x). 3) f(x) =7x2 + 5x - 9 Objective: (5.3) Find Second Derivative 4) f(x) =3x3 - 4x2 + 5 Objective: (5.3) Find Second Derivative 5) f(x) = 6x4 - 7x2 +7 Objective: (5.3) Find Second Derivative Use the Test for Concavity to determine where the given function is concave up and where it is concave down. Also find all inflection points. 6) f(x) =-2x3 - 18x2 - 4x + 7 Find the Possible Inflection Points and use them to find the endpoints of the Test Intervals. Use this table to help with the Concavity Test: Test Intervals Test Points (x) f" (x) Sign of f" (x) CC Up U / CC Down n Inflection Points CC Up Intervals: CC Down Intervals: Inflection Points: Objective: (5.3) Determine Concavity from Equation MATH 1325 - Test 3 Review 3 Dave RiceAnswer Key Testname: 1325 TEST 3 REVIEW 1) f(x) = 3x4 - 4x3 - 36x2 + 12 Find the Critical Numbers and use them to find the endpoints of the Test Intervals. f'(x) =12x3 - 12x2 - 72x = Set f'(x) = 0 = 12x3 - 12x2 - 72x = 0 = 12x(x2 - x-6) = 0 = 12x(x + 2)(x - 3) = 0= 12x = 0, x + 2 =0, x - 3 =0 = x = 0, x=-2, x = 3 (CNs) Use this table to help with the First Derivative Test: Test Intervals (-00, - 2) (-2, 0) (0, 3) (3, 00) Test Points (x) -3 -1 4 f' (x) = f'(-3) =-216 f'(-1) = 48 f'(1) = -72 f'(4) = 288 12x3 - 12x2- 72x Sign of f' (x) + + Incr 71 / Decr Local Extrema min max min (-2, f(-2)) = (-2, -52) (0, f(0)) = (0, 12) (3, f(3)) = (3, -177) Increasing Intervals: (-2, 0), (3, 0.) Decreasing Intervals: (-00, -2), (0, 3) Local Minima: (-2, -52), (3, -177) Local Maxima: (0, 12) MATH 1325 - Test 3 Review 7 Dave RiceAnswer Key Testname: 1325 TEST 3 REVIEW 8) f(x) = 3x4 - 8x3 - 18x2 + 10 f'(x) = 12x3 - 24x2 - 36x Find the Critical Numbers (CNs): f'(x) =0 = 12x3 - 24x2 - 36x = 0 = 12x(x2 - 2x - 3) = 0 = 12x(x + 1)(x - 3) = 0 = 12x = 0, x + 1 = 0, x - 3 = 0 = x = 0, x = -1, x = 3 = CNs: x = -1, 0, 3 f"(x) = 36x2 - 48x - 36 Plug the CNs into f"(x) and perform the Second Derivative Test: x =-1: f"(-1) =36(-1)2 -48(-1) - 36 = 48 >0 = minimum at x =-1 x = 0: f'(0) = 36(0)2 -48(0) - 36 = -36 0 = minimum at x = 3 Relavtive (Local) Minima in (x,y) format: (-1, f(-1)) = (-1, 3), (3, f(3)) = (3, -125) Relavtive (Local) Maxima in (x,y) format: (0, f(0)) = (0, 10) 9) f(x) = -x3 +3x2 + 72x + 100 f' (x) =-3x2 + 6x+72 Find the Critical Numbers (CNs): f'(x) =0 = -3x2 + 6x+ 72 = 0 = -3(x2 -2x - 24) = 0 = -3(x + 4)(x - 6) = 0 = x+4=0, x - 6 = 0 = X= -4, x =6 = CNs: x = -4, 6 f"(x) = -6x + 6 Plug the CNs into f"(x) and perform the Second Derivative Test: x =-4: f'(-4) = -6(-4) + 6 =30 >0 = minimum at x = 4 x = 6: f'(6) = -6(6) + 6 = -30