Explanation The given function is f ( x ) = 1 3 x 3 − 1 2 x 2 − 6 x + 3 and interval is [ − 4 , 4 ] . Find the critical number in the interval [ − 4 , 4 ] to compute the absolute minimum and absolute maximum. Differentiate the function f ( x ) = 1 3 x 3 − 1 2 x 2 − 6 x + 3 as follows, f ′ ( x ) = d d x ( 1 3 x 3 − 1 2 x 2 − 6 x + 3 ) = 1 3 ( 3 ) x 3 − 1 − 1 2 ( 2 ) x 2 − 1 − 6 + 0 = x 2 − x − 6 = ( x + 2 ) ( x − 3 ) Set f ′ ( x ) = 0 and solve for x . That is, x + 2 = 0 or x − 3 = 0 . Hence, the critical numbers are x = − 2 and x = 3 . Find the values of the function at the critical points and the end points of the given interval. Substitute x = − 4 in f ( x ) = 1 3 x 3 − 1 2 x 2 − 6 x + 3 , f ( − 4 ) = 1 3 ( − 4 ) 3 − 1 2 ( − 4 ) 2 − 6 ( − 4 ) + 3 = 1 3 ( − 64 ) − 1 2 ( 16 ) + 24 + 3 = − 21 .3333 − 8 − 24 + 3 ≈ − 2.3 That is, f ( − 4 ) ≈ − 2.33 . Substitute x = − 2 in f ( x ) = 1 3 x 3 − 1 2 x 2 − 6 x + 3 , f ( − 2 ) = 1 3 ( − 2 ) 3 − 1 2 ( − 2 ) 2 − 6 ( − 2 ) + 3 = 1 3 ( − 8 ) − 1 2 ( 4 ) + 12 + 3 = − 2

Finite Mathematics and Calculus with Applications (10th Edition)

10th Edition

ISBN: 9780321979407

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

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 14.1, Problem 14E

To determine

To find: The absolute extrema of f(x)=13x3−12x2−6x+3 on the domain [−4,4] if exist and verify the answer using the graphing calculator.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.1, Problem 13E

Chapter 14.1, Problem 15E

Students have asked these similar questions

An open-top rectangular box is being constructed to hold a volume of 150 in³. The base of the box is made from a material costing 7 cents/in². The front of the box must be decorated, and will cost 11 cents/in². The remainder of the sides will cost 3 cents/in². Find the dimensions that will minimize the cost of constructing this box. Please show your answers to at least 4 decimal places. Front width: Depth: in. in. Height: in.

Find and classify the critical points of z = (x² – 8x) (y² – 6y). Local maximums: Local minimums: Saddle points: - For each classification, enter a list of ordered pairs (x, y) where the max/min/saddle occurs. Enter DNE if there are no points for a classification.

Suppose that f(x, y, z) = (x − 2)² + (y – 2)² + (z − 2)² with 0 < x, y, z and x+y+z≤ 10. 1. The critical point of f(x, y, z) is at (a, b, c). Then a = b = C = 2. Absolute minimum of f(x, y, z) is and the absolute maximum is

Chapter 14 Solutions

Finite Mathematics and Calculus with Applications (10th Edition)

Ch. 14.1 - Find the absolute extrema of the function f(x) =...Ch. 14.1 - Find the locations and values of the absolute...Ch. 14.1 - Prob. 1WE Ch. 14.1 - Find all critical numbers for each of the...Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Find the locations of any absolute extrema for the...Ch. 14.1 - Prob. 5E Ch. 14.1 - Prob. 6E

Ch. 14.1 - Prob. 7E Ch. 14.1 - Prob. 8E Ch. 14.1 - Prob. 9E Ch. 14.1 - Prob. 10E Ch. 14.1 - Prob. 11E Ch. 14.1 - Prob. 12E Ch. 14.1 - Prob. 13E Ch. 14.1 - Prob. 14E Ch. 14.1 - Prob. 15E Ch. 14.1 - Prob. 16E Ch. 14.1 - Find the absolute extrema if they exist, as well...Ch. 14.1 - Prob. 18E Ch. 14.1 - Prob. 19E Ch. 14.1 - Prob. 20E Ch. 14.1 - Prob. 21E Ch. 14.1 - Prob. 22E Ch. 14.1 - Prob. 23E Ch. 14.1 - Prob. 24E Ch. 14.1 - Prob. 25E Ch. 14.1 - Prob. 26E Ch. 14.1 - Prob. 27E Ch. 14.1 - Prob. 28E Ch. 14.1 - Prob. 29E Ch. 14.1 - Prob. 30E Ch. 14.1 - Prob. 31E Ch. 14.1 - Prob. 32E Ch. 14.1 - Prob. 33E Ch. 14.1 - Prob. 34E Ch. 14.1 - Prob. 35E Ch. 14.1 - Find the absolute extrema if they exist, as well...Ch. 14.1 - Prob. 37E Ch. 14.1 - Prob. 38E Ch. 14.1 - Prob. 39E Ch. 14.1 - Prob. 40E Ch. 14.1 - Prob. 41E Ch. 14.1 - Prob. 42E Ch. 14.1 - Prob. 43E Ch. 14.1 - Prob. 44E Ch. 14.1 - Prob. 45E Ch. 14.1 - Prob. 46E Ch. 14.1 - Prob. 47E Ch. 14.1 - Prob. 48E Ch. 14.1 - Prob. 49E Ch. 14.1 - Prob. 50E Ch. 14.1 - Prob. 51E Ch. 14.1 - Prob. 53E Ch. 14.1 - Salmon Spawning The number of salmon swimming...Ch. 14.1 - Prob. 55E Ch. 14.1 - Prob. 56E Ch. 14.1 - Prob. 57E Ch. 14.1 - Prob. 58E Ch. 14.1 - Prob. 59E Ch. 14.1 - Prob. 60E Ch. 14.1 - Prob. 61E Ch. 14.1 - Prob. 62E Ch. 14.2 - Prob. 1YT Ch. 14.2 - Prob. 2YT Ch. 14.2 - Prob. 3YT Ch. 14.2 - Prob. 4YT Ch. 14.2 - Prob. 1WE Ch. 14.2 - Prob. 2WE Ch. 14.2 - Prob. 1E Ch. 14.2 - Prob. 2E Ch. 14.2 - Prob. 3E Ch. 14.2 - Prob. 4E Ch. 14.2 - Prob. 5E Ch. 14.2 - Prob. 6E Ch. 14.2 - Prob. 7E Ch. 14.2 - Prob. 8E Ch. 14.2 - Prob. 9E Ch. 14.2 - Prob. 10E Ch. 14.2 - Prob. 11E Ch. 14.2 - Prob. 12E Ch. 14.2 - Prob. 13E Ch. 14.2 - Prob. 14E Ch. 14.2 - Revenue A local club is arranging a charter flight...Ch. 14.2 - Prob. 16E Ch. 14.2 - Prob. 17E Ch. 14.2 - Prob. 18E Ch. 14.2 - Prob. 19E Ch. 14.2 - Prob. 20E Ch. 14.2 - Prob. 21E Ch. 14.2 - Prob. 22E Ch. 14.2 - Prob. 23E Ch. 14.2 - Prob. 24E Ch. 14.2 - Prob. 25E Ch. 14.2 - Container Design Your company needs to design...Ch. 14.2 - Prob. 27E Ch. 14.2 - Prob. 28E Ch. 14.2 - Prob. 29E Ch. 14.2 - Prob. 30E Ch. 14.2 - Prob. 31E Ch. 14.2 - Prob. 32E Ch. 14.2 - Prob. 33E Ch. 14.2 - Prob. 34E Ch. 14.2 - Prob. 35E Ch. 14.2 - Prob. 36E Ch. 14.2 - Prob. 37E Ch. 14.2 - Prob. 38E Ch. 14.2 - Prob. 39E Ch. 14.2 - Prob. 40E Ch. 14.2 - Maximum Sustainable Harvest In Exercises 41 and...Ch. 14.2 - Prob. 42E Ch. 14.2 - Prob. 43E Ch. 14.2 - Prob. 44E Ch. 14.2 - Harvesting Cod A recent article described the...Ch. 14.2 - Prob. 46E Ch. 14.2 - Prob. 47E Ch. 14.2 - Bird Migration Suppose a migrating bird flies at a...Ch. 14.2 - Prob. 49E Ch. 14.2 - Prob. 50E Ch. 14.2 - Prob. 51E Ch. 14.3 - Suppose the annual demand in Example 1 is only...Ch. 14.3 - Prob. 2YT Ch. 14.3 - Prob. 3YT Ch. 14.3 - Prob. 4YT Ch. 14.3 - Prob. 5YT Ch. 14.3 - Prob. 1WE Ch. 14.3 - Prob. 2WE Ch. 14.3 - In the discussion of economic lot size, use the...Ch. 14.3 - Prob. 2E Ch. 14.3 - Prob. 3E Ch. 14.3 - Prob. 4E Ch. 14.3 - Prob. 5E Ch. 14.3 - Prob. 6E Ch. 14.3 - Prob. 7E Ch. 14.3 - Prob. 8E Ch. 14.3 - Prob. 9E Ch. 14.3 - Prob. 10E Ch. 14.3 - Prob. 11E Ch. 14.3 - Prob. 12E Ch. 14.3 - Prob. 13E Ch. 14.3 - Prob. 14E Ch. 14.3 - Prob. 15E Ch. 14.3 - Prob. 16E Ch. 14.3 - Prob. 17E Ch. 14.3 - Prob. 18E Ch. 14.3 - Elasticity For each of the following demand...Ch. 14.3 - Elasticity For each of the following demand...Ch. 14.3 - Prob. 21E Ch. 14.3 - Elasticity For each of the following demand...Ch. 14.3 - Prob. 23E Ch. 14.3 - Prob. 24E Ch. 14.3 - Prob. 25E Ch. 14.3 - Prob. 26E Ch. 14.3 - Prob. 27E Ch. 14.3 - Prob. 28E Ch. 14.3 - Elasticity of Software The Valve Corporation, a...Ch. 14.3 - Prob. 30E Ch. 14.3 - Elasticity A study of the demand for air travel in...Ch. 14.3 - Elasticity The price along the West Coast of the...Ch. 14.3 - Prob. 33E Ch. 14.3 - Prob. 34E Ch. 14.4 - Prob. 1YT Ch. 14.4 - Prob. 2YT Ch. 14.4 - The graph of y4 x4 y2 + x2 = 0 is called the...Ch. 14.4 - Prob. 4YT Ch. 14.4 - Prob. 1WE Ch. 14.4 - Prob. 2WE Ch. 14.4 - Find dy/dx by implicit differentiation for the...Ch. 14.4 - Prob. 2E Ch. 14.4 - Prob. 3E Ch. 14.4 - Prob. 4E Ch. 14.4 - Prob. 5E Ch. 14.4 - Prob. 6E Ch. 14.4 - Prob. 7E Ch. 14.4 - Prob. 8E Ch. 14.4 - Prob. 9E Ch. 14.4 - Prob. 10E Ch. 14.4 - Prob. 11E Ch. 14.4 - Prob. 12E Ch. 14.4 - Prob. 13E Ch. 14.4 - Prob. 14E Ch. 14.4 - Prob. 15E Ch. 14.4 - Prob. 16E Ch. 14.4 - Prob. 17E Ch. 14.4 - Prob. 18E Ch. 14.4 - Prob. 19E Ch. 14.4 - Prob. 20E Ch. 14.4 - Prob. 21E Ch. 14.4 - Prob. 22E Ch. 14.4 - Prob. 23E Ch. 14.4 - Prob. 24E Ch. 14.4 - Prob. 25E Ch. 14.4 - Prob. 26E Ch. 14.4 - Prob. 27E Ch. 14.4 - Prob. 28E Ch. 14.4 - Prob. 29E Ch. 14.4 - Prob. 30E Ch. 14.4 - Prob. 31E Ch. 14.4 - Prob. 32E Ch. 14.4 - Prob. 33E Ch. 14.4 - Prob. 34E Ch. 14.4 - Prob. 35E Ch. 14.4 - Prob. 36E Ch. 14.4 - Prob. 37E Ch. 14.4 - Prob. 38E Ch. 14.4 - Prob. 39E Ch. 14.4 - Prob. 40E Ch. 14.4 - Prob. 41E Ch. 14.4 - Prob. 42E Ch. 14.4 - Prob. 43E Ch. 14.4 - Demand The demand equation for a certain product...Ch. 14.4 - Prob. 45E Ch. 14.4 - Prob. 46E Ch. 14.4 - Prob. 47E Ch. 14.4 - Prob. 48E Ch. 14.4 - Prob. 49E Ch. 14.4 - Prob. 50E Ch. 14.4 - Prob. 51E Ch. 14.4 - Prob. 52E Ch. 14.4 - Prob. 53E Ch. 14.5 - Prob. 1YT Ch. 14.5 - Prob. 2YT Ch. 14.5 - Prob. 3YT Ch. 14.5 - Prob. 4YT Ch. 14.5 - Prob. 1WE Ch. 14.5 - Prob. 2WE Ch. 14.5 - Prob. 1E Ch. 14.5 - Prob. 2E Ch. 14.5 - Prob. 3E Ch. 14.5 - Prob. 4E Ch. 14.5 - Prob. 5E Ch. 14.5 - Prob. 6E Ch. 14.5 - Prob. 7E Ch. 14.5 - Prob. 8E Ch. 14.5 - Prob. 9E Ch. 14.5 - Prob. 10E Ch. 14.5 - Prob. 11E Ch. 14.5 - Prob. 12E Ch. 14.5 - Prob. 13E Ch. 14.5 - Prob. 14E Ch. 14.5 - Prob. 15E Ch. 14.5 - Prob. 16E Ch. 14.5 - Prob. 17E Ch. 14.5 - Prob. 18E Ch. 14.5 - Prob. 19E Ch. 14.5 - Prob. 20E Ch. 14.5 - Prob. 21E Ch. 14.5 - Prob. 22E Ch. 14.5 - Prob. 23E Ch. 14.5 - Prob. 24E Ch. 14.5 - Prob. 25E Ch. 14.5 - Prob. 26E Ch. 14.5 - Prob. 27E Ch. 14.5 - Prob. 28E Ch. 14.5 - Prob. 29E Ch. 14.5 - Prob. 30E Ch. 14.5 - Prob. 31E Ch. 14.5 - Prob. 32E Ch. 14.6 - Prob. 1YT Ch. 14.6 - Prob. 2YT Ch. 14.6 - Prob. 3YT Ch. 14.6 - Prob. 1WE Ch. 14.6 - Prob. 2WE Ch. 14.6 - Prob. 1E Ch. 14.6 - Prob. 2E Ch. 14.6 - Prob. 3E Ch. 14.6 - Prob. 4E Ch. 14.6 - Prob. 5E Ch. 14.6 - For Exercises 18, find dy for the given values of...Ch. 14.6 - For Exercises 18, find dy for the given values of...Ch. 14.6 - Prob. 8E Ch. 14.6 - Prob. 9E Ch. 14.6 - Prob. 10E Ch. 14.6 - Prob. 11E Ch. 14.6 - Prob. 12E Ch. 14.6 - Prob. 13E Ch. 14.6 - Prob. 14E Ch. 14.6 - Use the differential to approximate each quantity....Ch. 14.6 - Prob. 16E Ch. 14.6 - Business and Economies 17.Demand The demand for...Ch. 14.6 - Prob. 18E Ch. 14.6 - Revenue A company estimates that the revenue (in...Ch. 14.6 - Prob. 20E Ch. 14.6 - Prob. 21E Ch. 14.6 - Prob. 22E Ch. 14.6 - Prob. 23E Ch. 14.6 - Prob. 24E Ch. 14.6 - Prob. 25E Ch. 14.6 - Prob. 26E Ch. 14.6 - Prob. 27E Ch. 14.6 - Prob. 28E Ch. 14.6 - Prob. 29E Ch. 14.6 - Gray Wolves Accurate methods of estimating the age...Ch. 14.6 - Prob. 31E Ch. 14.6 - Prob. 32E Ch. 14.6 - Prob. 33E Ch. 14.6 - Prob. 34E Ch. 14.6 - Prob. 35E Ch. 14.6 - Prob. 36E Ch. 14.6 - Prob. 37E Ch. 14.6 - Prob. 38E Ch. 14.6 - Prob. 39E Ch. 14.6 - Prob. 40E Ch. 14.6 - Prob. 41E Ch. 14 - The absolute maximum of a function always occurs...Ch. 14 - A continuous function on a closed interval has an...Ch. 14 - Prob. 3RE Ch. 14 - Prob. 4RE Ch. 14 - Total revenue is maximized at the price where...Ch. 14 - Prob. 6RE Ch. 14 - Prob. 7RE Ch. 14 - Prob. 8RE Ch. 14 - Prob. 9RE Ch. 14 - Prob. 10RE Ch. 14 - Prob. 11RE Ch. 14 - Prob. 12RE Ch. 14 - Prob. 13RE Ch. 14 - Prob. 14RE Ch. 14 - Prob. 15RE Ch. 14 - Prob. 16RE Ch. 14 - Prob. 17RE Ch. 14 - Prob. 18RE Ch. 14 - Prob. 19RE Ch. 14 - Prob. 20RE Ch. 14 - Prob. 21RE Ch. 14 - Prob. 22RE Ch. 14 - Prob. 23RE Ch. 14 - Prob. 24RE Ch. 14 - Prob. 25RE Ch. 14 - Prob. 26RE Ch. 14 - Prob. 27RE Ch. 14 - Prob. 28RE Ch. 14 - Prob. 29RE Ch. 14 - Prob. 30RE Ch. 14 - Prob. 31RE Ch. 14 - Why is implicit differentiation used in related...Ch. 14 - Prob. 33RE Ch. 14 - Prob. 34RE Ch. 14 - Prob. 35RE Ch. 14 - Prob. 36RE Ch. 14 - Prob. 37RE Ch. 14 - Prob. 38RE Ch. 14 - Prob. 39RE Ch. 14 - Prob. 40RE Ch. 14 - Prob. 41RE Ch. 14 - Prob. 42RE Ch. 14 - Prob. 43RE Ch. 14 - Prob. 44RE Ch. 14 - Prob. 45RE Ch. 14 - Prob. 46RE Ch. 14 - Prob. 47RE Ch. 14 - Prob. 48RE Ch. 14 - Prob. 49RE Ch. 14 - Prob. 50RE Ch. 14 - Prob. 51RE Ch. 14 - Prob. 52RE Ch. 14 - Prob. 53RE Ch. 14 - Prob. 54RE Ch. 14 - Prob. 55RE Ch. 14 - Prob. 56RE Ch. 14 - Prob. 57RE Ch. 14 - Prob. 58RE Ch. 14 - Prob. 59RE Ch. 14 - Prob. 60RE Ch. 14 - Prob. 61RE Ch. 14 - Prob. 62RE Ch. 14 - Prob. 63RE Ch. 14 - Prob. 64RE Ch. 14 - Prob. 65RE Ch. 14 - Prob. 66RE Ch. 14 - Prob. 67RE Ch. 14 - Prob. 68RE

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 extrema of f ( x ) = 1 3 x 3 − 1 2 x 2 − 6 x + 3 on the domain [ − 4 , 4 ] if exist and verify the answer using the graphing calculator.

Solution Summary: The author explains how to find the absolute extrema of f(x)=13x

Concept explainers

Videos

To find: The absolute extrema of f(x)=13x3−12x2−6x+3 on the domain [−4,4] if exist and verify the answer using the graphing calculator.

Want to see the full answer?

Chapter 14 Solutions