Explanation Given Information: The provided function is: f ( x ) = 1 3 x 3 − x 2 − 3 x + 5 and the interval is [ − 2 , 0 ] . Formula used: Maximum-Minimum Principle: Suppose f is a function such that f ′ ( x ) exists for every x in an interval I and there is exactly one ( critical ) value c in I , for which f ′ ( c ) = 0 . Then f ( c ) is the absolute maximum value over I if f ″ ( c ) < 0 or f ( c ) is the absolute minimum value over I if f ″ ( c ) > 0 . Calculation: Consider the provided function: f ( x ) = 1 3 x 3 − x 2 − 3 x + 5 with interval [ − 2 , 0 ] Differentiate f ( x ) = 1 3 x 3 − x 2 − 3 x + 5 with respect to x : d d x f ( x ) = d d x ( 1 3 x 3 − x 2 − 3 x + 5 ) f ′ ( x ) = 1 3 d ( x 3 ) d x − d ( x 2 ) d x − 3 d x d x + d ( 5 ) d x = 1 3 ⋅ 3 ( x 3 − 1 ) − 2 x 2 − 1 − 3 ( 1 ) + 0 = x 2 − 2 x − 3 Substitute f ′ ( x ) = 0 in the above expression: x 2 − 2 x − 3 = 0 ( x − 3 ) ( x + 1 ) = 0 Apply zero product rule on above factors: ( x − 3 ) = 0 or ( x + 1 ) = 0 If ( x − 3 ) = 0 : ( x − 3 ) = 0 x − 3 = 0 x = 3 If ( x + 1 ) = 0 : ( x + 1 ) = 0 x + 1 = 0 x = − 1 For Maximum − Minimum Principal, differentiate f ′ ( x ) = x 2 − 2 x − 3 with respect to x : f ″ ( x ) = d ( x 2 ) d x − 2 d x d x − d ( 3 ) d x = 2 x 2 − 1 − 2 ( 1 ) − 0 = 2 x − 2 As, all points are 3, − 1 and interval [ − 2 , 0 ] Here, 3 would be excluded

Calculus and Its Applications (11th Edition)

11th Edition

ISBN: 9780321979391

Author: Marvin L. Bittinger, David J. Ellenbogen, Scott J. Surgent

Publisher: PEARSON

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Chapter CR, Problem 37E

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To calculate: The absolute maximum and minimum value, if exist, of function f(x)=13x3−x2−3x+5 in interval [−2,0].

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Chapter CR, Problem 36E

Chapter CR, Problem 38E

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Chapter CR Solutions

Calculus and Its Applications (11th Edition)

Ch. CR - Write an equation of the line with slope 4 and...Ch. CR - Prob. 2E Ch. CR - For f(x)=x25, find f(x+h). x2+2xh+h25 Ch. CR - 4. a. Graph: b. Find. c. Find. d. Is f...Ch. CR - Prob. 5E Ch. CR - Find each limit, if it exists. If a limit does not...Ch. CR - Find each limit, if it exists. If a limit does not...Ch. CR - Find each limit, if it exists. If a limit does not...Ch. CR - Find each limit, if it exists. If a limit does not...Ch. CR - Find each limit, if it exists. If a limit does not...

Ch. CR - Find each limit, if it exists. If a limit does not...Ch. CR - For Exercises 12-14, refer to the following graph...Ch. CR - For Exercises 12-14, refer to the following graph...Ch. CR - For Exercises 12-14, refer to the following graph...Ch. CR - Differentiate. y=9x+3 Ch. CR - Differentiate. y=x27x+3 Ch. CR - Differentiate. y=x1/4 Ch. CR - Differentiate. f(x)=x6 Ch. CR - Prob. 19E Ch. CR - Differentiate. 22. Ch. CR - Prob. 21E Ch. CR - Differentiate. y=elnx Ch. CR - Prob. 23E Ch. CR - Differentiate. 24. Ch. CR - Differentiate. 25. Ch. CR - 26. For find. Ch. CR - Business: average cost. Doubletake Clothing finds...Ch. CR - 28. Differentiate implicitly to find if . Ch. CR - Find an equation of the tangent line to the graph...Ch. CR - 30. Find the x-value(s) at which the tangent line...Ch. CR - Sketch the graph of each function. List the label...Ch. CR - Prob. 32E Ch. CR - Sketch the graph of each function. List the label...Ch. CR - Prob. 34E Ch. CR - Find the absolute maximum and minimum values, if...Ch. CR - Prob. 36E Ch. CR - Prob. 37E Ch. CR - Prob. 38E Ch. CR - 39. Business: minimizing inventory costs. An...Ch. CR - Prob. 40E Ch. CR - Business: exponential growth. Friedas Frozen...Ch. CR - Prob. 42E Ch. CR - 43. Business: approximating cost average. A square...Ch. CR - Prob. 44E Ch. CR - Prob. 46E Ch. CR - Prob. 47E Ch. CR - Evaluate. 48. (Use Table 1 on pp. 431-432) Ch. CR - Prob. 49E Ch. CR - Evaluate. (x+3)lnxdx Ch. CR - Prob. 51E Ch. CR - Prob. 52E Ch. CR - 53. Find the area under the graph of over the...Ch. CR - Business: present value. Find the present value of...Ch. CR - Prob. 55E Ch. CR - Evaluate. 56. Business: contract buyout. An...Ch. CR - Prob. 57E Ch. CR - 58. Economic: supply and demand. Demand and supply...Ch. CR - 59. Find the volume of the solid of revolution...Ch. CR - 60. Find the volume of the solid of revolution...Ch. CR - Consider the data in the following table. Age of...Ch. CR - Prob. 62E Ch. CR - Given find each of the following. 63. Ch. CR - Prob. 64E Ch. CR - 65. Maximize subject to the constraint. Ch. CR - 66. Evaluate . Ch. CR - Prob. 67E Ch. CR - Solve the differential equation dy/dx=xy.Ch. CR - Solve the differential equation y+4xy=3x, where...Ch. CR - Prob. 70E Ch. CR - Prob. 71E Ch. CR - Business: distribution of weights. The weight, in...Ch. CR - Business: wait times. The wait time t in minutes,...Ch. CR - Prob. 74E Ch. CR - 75. Business: distribution of salaries. The...

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The absolute maximum and minimum value, if exist, of function f ( x ) = 1 3 x 3 − x 2 − 3 x + 5 in interval [ − 2 , 0 ] .

Solution Summary: The author calculates the absolute maximum and minimum value of the function f(x) = 13x