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

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter CR, Problem 37E

To determine

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter CR, Problem 36E

Chapter CR, Problem 38E

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...

Additional Math Textbook Solutions

Find more solutions based on key concepts

Retirement Income Several times during the year, the U.S. Census Bureau takes random samples from the populatio...

Introductory Statistics

In Problems 13-22, for the given functions f and g , find: a. ( fg )( 4 ) b. ( gf )( 2 ) c. ( ff )( 1 ) d. ( gg...

Precalculus

Fill in each blank so that the resulting statement is true. The quadratic function f(x)=a(xh)2+k,a0, is in ____...

Algebra and Trigonometry (6th Edition)

Hypothesis Testing Using a P-Value In Exercises 31–36, identify the claim and state H0 and Ha. find the standar...

Elementary Statistics: Picturing the World (7th Edition)

Evaluate the integrals in Exercises 1–10. 7.

University Calculus: Early Transcendentals (4th Edition)

The equivalent expression of x(y+z) by using the commutative property.

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Similar questions

Find all real values of x for which fx = 0, where fx=x216.
Find all real zeros of fx=8x34x2+6x3.
Find the domain of the function f(x)=142xx2

Recommended textbooks for you

Algebra: Structure And Method, Book 1
Algebra
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:McDougal Littell
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
College Algebra
Algebra
ISBN:9781337282291
Author:Ron Larson
Publisher:Cengage Learning
Intermediate Algebra
Algebra
ISBN:9780998625720
Author:Lynn Marecek
Publisher:OpenStax College
Big Ideas Math A Bridge To Success Algebra 1: Stu...
Algebra
ISBN:9781680331141
Author:HOUGHTON MIFFLIN HARCOURT
Publisher:Houghton Mifflin Harcourt
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage

Algebra: Structure And Method, Book 1

Algebra

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:McDougal Littell

College Algebra (MindTap Course List)

Algebra

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:Cengage Learning

College Algebra

Algebra

ISBN:9781337282291

Author:Ron Larson

Publisher:Cengage Learning

Intermediate Algebra

Algebra

ISBN:9780998625720

Author:Lynn Marecek

Publisher:OpenStax College

Big Ideas Math A Bridge To Success Algebra 1: Stu...

Algebra

ISBN:9781680331141

Author:HOUGHTON MIFFLIN HARCOURT

Publisher:Houghton Mifflin Harcourt

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:9781133382119

Author:Swokowski

Publisher:Cengage

SEE MORE TEXTBOOKS

Finding Local Maxima and Minima by Differentiation; Author: Professor Dave Explains;https://www.youtube.com/watch?v=pvLj1s7SOtk;License: Standard YouTube License, CC-BY

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

Concept explainers

Videos

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

Want to see the full answer?

Chapter CR Solutions

Additional Math Textbook Solutions