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

16th Edition

ISBN: 9781323161470

Author: BITTINGER

Publisher: PEARSON C

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

Students have asked these similar questions

Only 100% sure experts solve it correct complete solutions ok

rmine the immediate settlement for points A and B shown in figure below knowing that Aq,-200kN/m², E-20000kN/m², u=0.5, Depth of foundation (DF-0), thickness of layer below footing (H)=20m. 4m B 2m 2m A 2m + 2m 4m

sy = f(x) + + + + + + + + + X 3 4 5 7 8 9 The function of shown in the figure is continuous on the closed interval [0, 9] and differentiable on the open interval (0, 9). Which of the following points satisfies conclusions of both the Intermediate Value Theorem and the Mean Value Theorem for f on the closed interval [0, 9] ? (A A B B C D

Chapter CR Solutions

CALCULUS & ITS APPLICATIONS+MYMATHLAB

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.

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