12 Use the given graph of to find the following. (a) The open intervals on which is increasing. (b) The open intervals on which is decreasing. (c) The open intervals on which is concave upward. (d) The open intervals on which is concave downward. (e) The coordinates of the points of inflection.
Read moreTable of Contents
1.1
Four Ways to Represent a Function
1.2
Mathematical Models: A Catalog of Essential Functions
1.3
New Functions from Old Functions
1.4
Graphing Calculators and Computers
1.5
Exponential Functions
1.6
Inverse Functions and Logarithms
2.1
The Tangent and Velocity Problems
2.2
The Limit of a Function
2.3
Calculating Limits Using the Limit Laws
2.4
The Precise Definition of a Limit
2.5
Continuity
2.6
Limits at Infinity; Horizontal Asymptotes
2.7
Derivatives and Rates of Change
2.8
The Derivative as a Function
3.1
Derivatives of Polynomials and Exponential Functions
3.10
Linear Approximations and Differentials
3.11
Hyperbolic Functions
3.2
The Product and Quotient Rules
3.3
Derivatives of Trigonometric Functions
3.4
The Chain Rule
3.5
Implicit Differentiation
3.6
Derivatives of Logarithmic Functions
3.7
Rates of Change in the Natural and Social Sciences
3.8
Exponential Growth and Decay
3.9
RELATED RATES
4.1
Maximum and Minimum Values
4.2
The Mean Value Theorem
4.3
How Derivatives Affect the Shape of a Graph
4.4
Indeterminate Forms and lHospitals Rule
4.5
Summary of Curve Sketching
4.6
Graphing with Calculus and Calculators
4.7
Optimization Problems
4.8
Newtons Method
4.9
Antiderivatives
5.1
Areas and Distances
5.2
The Definite Integral
5.3
The Fundamental Theorem of Calculus
5.4
Indefinite Integrals and the Net Change Theorem
6.1
Areas Between Curves
6.2
Volumes
6.3
Volumes by Cylindrical Shells
6.4
Work
6.5
Average Value of a Function
7.1
Integration by Parts
7.2
Trigonometric Integrals
7.3
Trigonometric Substitution
7.4
Integration of Rational Functions by Partial Fractions
7.5
Strategy for Integration
7.6
Integration Using Tables and Computer Algebra Systems
7.7
Approximate Integration
7.8
Improper Integrals
8.1
Arc Length
8.2
Area of a Surface of Revolution
8.3
Applications to Physics and Engineering
8.4
Applications to Economics and Biology
8.5
Probability
9.1
Modeling with Differential Equations
9.2
Direction Fields and Eulers Method
9.3
Separable Equations
9.4
Models for Population Growth
9.5
Linear Equations
9.6
Predator-Prey Systems
10.1
Curves Defined by Parametric Equations
10.2
Calculus with Parametric Curves
10.3
Polar Coordinates
10.4
Areas and Lengths in Polar Coordinates
10.5
Conic Sections
10.6
Conic Sections in Polar Coordinates
11.1
Sequences
11.10
Taylor and Maclaurin Series
11.11
Applications of Taylor Polynomials
11.2
Series
11.3
The Integral Test and Estimates of Sums
11.4
The Comparison Tests
11.5
Alternating Series
11.6
Absolute Convergence and the Ratio and Root Tests
11.7
Strategy for Testing Series
11.8
Power Series
11.9
Representations of Functions as Power Series
12.1
Three-Dimensional Coordinate Systems
12.2
Vectors
12.3
The Dot Product
12.4
The Cross Product
12.5
Equations of Lines and Planes
12.6
Cylinders and Quadric Surfaces
13.1
Vector Functions
13.2
Derivatives and Integrals of Vector Functions
13.3
Arc Length and Curvature
13.4
Motion in Space: Velocity and Acceleration
14.1
Functions of Several Variables
14.2
Limits and Continuity
14.3
Partial Derivatives
14.4
Tangent Planes and Linear Approximations
14.5
The Chain Rule
14.6
Directional Derivatives and the Gradient Vector
14.7
Maximum and Minimum Values
14.8
Lagrange Multipliers
15.1
Double Integrals over Rectangles
15.10
Change of Variables in Multiple Integrals
15.2
Iterated Integrals
15.3
Double Integrals over General Regions
15.4
Double Integrals in Polar Coordinates
15.5
Applications of Double Integrals
15.6
Surface Area
15.7
Triple Integrals
15.8
Triple Integrals in Cylindrical Coordinates
15.9
Triple Integrals in Spherical Coordinates
16.1
Vector Fields
16.10
Summary
16.2
Line Integrals
16.3
The Fundamental Theorem for Line Integrals
16.4
Greens Theorem
16.5
Curl and Divergence
16.6
Parametric Surfaces and Their Areas
16.7
Surface Integrals
16.8
Stokes Theorem
16.9
The Divergence Theorem
17.1
Second-Order Linear Equations
17.2
Nonhomogeneous Linear Equations
17.3
Applications of Second-Order Differential Equations
17.4
Series Solutions
Textbook Solutions for Calculus: Early Transcendentals
Chapter 4.3 Problem 76
Question
7476 Assume that all of the functions are twice differentiable and the second derivatives are never 0.Suppose and are both concave upward on .Under what condition on will the composite functionbe concave upward?
Solution
The first step in solving 4.3 problem number 76 trying to solve the problem we have to refer to the textbook question: 7476 Assume that all of the functions are twice differentiable and the second derivatives are never 0.Suppose and are both concave upward on .Under what condition on will the composite functionbe concave upward?
From the textbook chapter How Derivatives Affect the Shape of a Graph you will find a few key concepts needed to solve this.
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full solution
Title
Calculus: Early Transcendentals 7
Author
ames Stewart
ISBN
9780538497909
Solved: 7476 Assume that all of the functions are twice differentiable and the second
Chapter 4.3 textbook questions
-
Chapter 4: Problem 1 Calculus: Early Transcendentals 7
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Chapter 4: Problem 2 Calculus: Early Transcendentals 7
12 Use the given graph of to find the following. (a) The open intervals on which is increasing. (b) The open intervals on which is decreasing. (c) The open intervals on which is concave upward. (d) The open intervals on which is concave downward. (e) The coordinates of the points of inflection.
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Chapter 4: Problem 3 Calculus: Early Transcendentals 7
Suppose you are given a formula for a function . (a) How do you determine where is increasing or decreasing?(b) How do you determine where the graph of is concave upward or concave downward? (c) How do you locate inflection points?
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Chapter 4: Problem 4 Calculus: Early Transcendentals 7
(a) State the First Derivative Test. (b) State the Second Derivative Test. Under what circum - stances is it inconclusive? What do you do if it fails?
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Chapter 4: Problem 5 Calculus: Early Transcendentals 7
56 The graph of the derivative of a function is shown. (a) On what intervals is increasing or decreasing? (b) At what values of x does have a local maximum or minimum?
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Chapter 4: Problem 6 Calculus: Early Transcendentals 7
56 The graph of the derivative of a function is shown. (a) On what intervals is increasing or decreasing? (b) At what values of x does have a local maximum or minimum?
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Chapter 4: Problem 7 Calculus: Early Transcendentals 7
7. In each part state the -coordinates of the inflection points of . Give reasons for your answers. (a) The curve is the graph of . (b) The curve is the graph of . (c) The curve is the graph of .
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Chapter 4: Problem 8 Calculus: Early Transcendentals 7
The graph of the first derivative of a function is shown. (a) On what intervals is increasing? Explain. (b) At what values of does have a local maximum or minimum? Explain. (c) On what intervals is concave upward or concave down ward? Explain. (d) What are the -coordinates of the inflection points of ? Why?
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Chapter 4: Problem 9 Calculus: Early Transcendentals 7
918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx 2x 3 3x 2 36x f
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Chapter 4: Problem 10 Calculus: Early Transcendentals 7
918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx 4x 3 3x 2 6x 1 fx
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Chapter 4: Problem 11 Calculus: Early Transcendentals 7
918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx x4 2x2 3 fx
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Chapter 4: Problem 12 Calculus: Early Transcendentals 7
918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx x x 2 1 fx
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Chapter 4: Problem 13 Calculus: Early Transcendentals 7
918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx sin x cos x, 0 x 2 fx
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Chapter 4: Problem 14 Calculus: Early Transcendentals 7
918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx cos2 x 2 sin x 0 x 2 fx
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Chapter 4: Problem 15 Calculus: Early Transcendentals 7
918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx e2x ex fx x
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Chapter 4: Problem 16 Calculus: Early Transcendentals 7
918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx x 2 ln x fx
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Chapter 4: Problem 17 Calculus: Early Transcendentals 7
918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx x 2 x ln x fx x
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Chapter 4: Problem 18 Calculus: Early Transcendentals 7
918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx x 4 ex fx
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Chapter 4: Problem 19 Calculus: Early Transcendentals 7
1921 Find the local maximum and minimum values of using both the First and Second Derivative Tests. Which method do you prefer fx 1 3x 2 2x 3 fx
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Chapter 4: Problem 20 Calculus: Early Transcendentals 7
1921 Find the local maximum and minimum values of using both the First and Second Derivative Tests. Which method do you preferfx x 2 x 1 f
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Chapter 4: Problem 21 Calculus: Early Transcendentals 7
1921 Find the local maximum and minimum values of using both the First and Second Derivative Tests. Which method do you preferfx sx s 4 x fx
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Chapter 4: Problem 22 Calculus: Early Transcendentals 7
. (a) Find the critical numbers of . (b) What does the Second Derivative Test tell you about the behavior of at these critical numbers? (c) What does the First Derivative Test tell you?
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Chapter 4: Problem 23 Calculus: Early Transcendentals 7
. Suppose is continuous on . (a) If and , what can you say about ? (b) If and , what can you say about ?
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Chapter 4: Problem 24 Calculus: Early Transcendentals 7
2429 Sketch the graph of a function that satisfies all of the given conditions.Vertical asymptote , if , if , if , if
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Chapter 4: Problem 25 Calculus: Early Transcendentals 7
2429 Sketch the graph of a function that satisfies all of the given conditions.f0 f2 f4 0 f x 0
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Chapter 4: Problem 26 Calculus: Early Transcendentals 7
2429 Sketch the graph of a function that satisfies all of the given conditions.f 1 f 1 0 f x 0 x 1 f x 0
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Chapter 4: Problem 27 Calculus: Early Transcendentals 7
2429 Sketch the graph of a function that satisfies all of the given conditions.f x 0 x 2 f x 0 x 2 f
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Chapter 4: Problem 28 Calculus: Early Transcendentals 7
2429 Sketch the graph of a function that satisfies all of the given conditions.f x 0 x 2 f x 0 x 2 f
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Chapter 4: Problem 29 Calculus: Early Transcendentals 7
2429 Sketch the graph of a function that satisfies all of the given conditions.fx 0 fx 0 x f
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Chapter 4: Problem 30 Calculus: Early Transcendentals 7
Suppose , and and for all . (a) Sketch a possible graph for . (b) How many solutions does the equation have? Why? (c) Is it possible that ? Why?
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Chapter 4: Problem 31 Calculus: Early Transcendentals 7
3132 The graph of the derivative of a continuous function is shown. (a) On what intervals is increasing? Decreasing? (b) At what values of x does have a local maximum? Local minimum? (c) On what intervals is concave upward? Concave downward? (d) State the x-coordinate(s) of the point(s) of inflection. (e) Assuming that , sketch a graph of f.
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Chapter 4: Problem 32 Calculus: Early Transcendentals 7
3132 The graph of the derivative of a continuous function is shown. (a) On what intervals is increasing? Decreasing? (b) At what values of x does have a local maximum? Local minimum? (c) On what intervals is concave upward? Concave downward? (d) State the x-coordinate(s) of the point(s) of inflection. (e) Assuming that , sketch a graph of f.
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Chapter 4: Problem 33 Calculus: Early Transcendentals 7
33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.fx x 3 12x 2 y
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Chapter 4: Problem 34 Calculus: Early Transcendentals 7
33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.fx 36x 3x 2 2x 3 fx
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Chapter 4: Problem 35 Calculus: Early Transcendentals 7
33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.fx 2 2x 2 x 4 fx
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Chapter 4: Problem 36 Calculus: Early Transcendentals 7
33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.tx 200 8x 3 x 4 fx
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Chapter 4: Problem 37 Calculus: Early Transcendentals 7
33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.hx x 1 5 5x 2 tx 20
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Chapter 4: Problem 38 Calculus: Early Transcendentals 7
33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.hx 5x 3 3x 5 hx
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Chapter 4: Problem 39 Calculus: Early Transcendentals 7
33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.Fx xs6 x hx
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Chapter 4: Problem 40 Calculus: Early Transcendentals 7
33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.Gx 5x 23 2x 53 Fx
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Chapter 4: Problem 42 Calculus: Early Transcendentals 7
33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.Cx x 4
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Chapter 4: Problem 43 Calculus: Early Transcendentals 7
33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.fx lnx Cx
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Chapter 4: Problem 44 Calculus: Early Transcendentals 7
33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.f 2 cos cos 0 2 2
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Chapter 4: Problem 46 Calculus: Early Transcendentals 7
4552 (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of .fx 1 1 x 1 x 2 f
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Chapter 4: Problem 47 Calculus: Early Transcendentals 7
4552 (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of .fx x 2 4 x 2 4 fx
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Chapter 4: Problem 48 Calculus: Early Transcendentals 7
4552 (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of .fx sx 2 1 x fx
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Chapter 4: Problem 49 Calculus: Early Transcendentals 7
4552 (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of .fx e x 1 e x fx s
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Chapter 4: Problem 50 Calculus: Early Transcendentals 7
4552 (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of .fx e ln
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Chapter 4: Problem 51 Calculus: Early Transcendentals 7
4552 (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of .fx ln1 ln x fx x
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Chapter 4: Problem 52 Calculus: Early Transcendentals 7
4552 (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of .fx earctan x fx
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Chapter 4: Problem 53 Calculus: Early Transcendentals 7
Suppose the derivative of a function is . On what interval is increasing?
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Chapter 4: Problem 54 Calculus: Early Transcendentals 7
Use the methods of this section to sketch the curve , where is a positive constant. What do the members of this family of curves have in common? How do they differ from each other?
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Chapter 4: Problem 55 Calculus: Early Transcendentals 7
5556 (a) Use a graph of to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of at which increases most rapidly. Then find the exact value.fx x 1 sx 2 1 fx
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Chapter 4: Problem 56 Calculus: Early Transcendentals 7
5556 (a) Use a graph of to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of at which increases most rapidly. Then find the exact value.f x x 2 ex fx
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Chapter 4: Problem 57 Calculus: Early Transcendentals 7
5758 (a) Use a graph of to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of to give better estimates.fx cos x 0 x 2 1
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Chapter 4: Problem 58 Calculus: Early Transcendentals 7
5758 (a) Use a graph of to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of to give better estimates.fx x 3 x 2 4 fx c
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Chapter 4: Problem 59 Calculus: Early Transcendentals 7
5960 Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph .fx x 4 x 3 1 sx 2 x 1 f
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Chapter 4: Problem 60 Calculus: Early Transcendentals 7
5960 Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph .fx x 2 tan1 x 1 x 3 fx
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Chapter 4: Problem 61 Calculus: Early Transcendentals 7
A graph of a population of yeast cells in a new laboratory culture as a function of time is shown. (a) Describe how the rate of population increase varies. (b) When is this rate highest? (c) On what intervals is the population function concave upward or downward? (d) Estimate the coordinates of the inflection point.
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Chapter 4: Problem 62 Calculus: Early Transcendentals 7
Let be the temperature at time where you live and suppose that at time you feel uncomfortably hot. How do you feel about the given data in each case?
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Chapter 4: Problem 63 Calculus: Early Transcendentals 7
Let be a measure of the knowledge you gain by studying for a test for t hours. Which do you think is larger, or ? Is the graph of K concave upward or concave downward? Why?
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Chapter 4: Problem 64 Calculus: Early Transcendentals 7
Coffee is being poured into the mug shown in the figure at a constant rate (measured in volume per unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time. Account for the shape of the graph in terms of concavity. What is the significance of the inflection point?
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Chapter 4: Problem 65 Calculus: Early Transcendentals 7
A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, , and is measured in minutes, estimate the times corresponding to the inflection points and explain their significance. If you have a graphing device, use it to graph the drug response curve.
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Chapter 4: Problem 66 Calculus: Early Transcendentals 7
The family of bell-shaped curves occurs in probability and statistics, where it is called the normal density function. The constant is called the mean and the positive constant is called the standard deviation. For simplicity, lets scale the function so as to remove the factor and lets analyze the special case where . So we study the function (a) Find the asymptote, maximum value, and inflection points of . (b) What role does play in the shape of the curve? ; (c) Illustrate by graphing four members of this family on the same screen.
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Chapter 4: Problem 67 Calculus: Early Transcendentals 7
. Find a cubic function that has a local maximum value of at and a local minimum value of 0 at .
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Chapter 4: Problem 68 Calculus: Early Transcendentals 7
For what values of the numbers and does the function have the maximum value ?
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Chapter 4: Problem 69 Calculus: Early Transcendentals 7
(a) If the function has the local minimum value at , what are the values of and ? (b) Which of the tangent lines to the curve in part (a) has the smallest slope?
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Chapter 4: Problem 70 Calculus: Early Transcendentals 7
For what values of and is an inflection point of the curve ? What additional inflection points does the curve have?
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Chapter 4: Problem 71 Calculus: Early Transcendentals 7
Show that the curve has three points of inflection and they all lie on one straight line.
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Chapter 4: Problem 72 Calculus: Early Transcendentals 7
Show that the curves and touch the curve at its inflection points.
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Chapter 4: Problem 73 Calculus: Early Transcendentals 7
Show that the inflection points of the curve lie on the curve .
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Chapter 4: Problem 74 Calculus: Early Transcendentals 7
7476 Assume that all of the functions are twice differentiable and the second derivatives are never 0.(a) If and are concave upward on , show that is concave upward on . (b) If is positive and concave upward on , show that the function is concave upward on .
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Chapter 4: Problem 75 Calculus: Early Transcendentals 7
7476 Assume that all of the functions are twice differentiable and the second derivatives are never 0.(a) If and are positive, increasing, concave upward functions on , show that the product function is concave upward on . (b) Show that part (a) remains true if and are both decreasing. (c) Suppose is increasing and is decreasing. Show, by giving three examples, that may be concave upward, concave downward, or linear. Why doesnt the argument in parts (a) and (b) work in this case?
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Chapter 4: Problem 76 Calculus: Early Transcendentals 7
7476 Assume that all of the functions are twice differentiable and the second derivatives are never 0.Suppose and are both concave upward on . Under what condition on will the composite function be concave upward?
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Chapter 4: Problem 77 Calculus: Early Transcendentals 7
Show that for . [Hint: Show that is increasing on .]
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Chapter 4: Problem 78 Calculus: Early Transcendentals 7
(a) Show that for . (b) Deduce that for . (c) Use mathematical induction to prove that for and any positive integer ,
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Chapter 4: Problem 79 Calculus: Early Transcendentals 7
Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three -intercepts , and , show that the -coordinate of the inflection point is . b x 1s3 a 2 9s3 f x x 3 ax 2 bx y 2 x 2 4 4x 2 y x sin x x1 x2 x3 3 x x1, x2 x
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Chapter 4: Problem 80 Calculus: Early Transcendentals 7
For what values of does the polynomial have two inflection points? One inflection point? None? Illustrate by graphing for several values of . How does the graph change as decreases?
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Chapter 4: Problem 81 Calculus: Early Transcendentals 7
Prove that if is a point of inflection of the graph of and exists in an open interval that contains , then . [Hint: Apply the First Derivative Test and Fermats Theorem to the function .]
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Chapter 4: Problem 82 Calculus: Early Transcendentals 7
Show that if , then , but is not an inflection point of the graph of .
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Chapter 4: Problem 83 Calculus: Early Transcendentals 7
Show that the function has an inflection point at but does not exist
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Chapter 4: Problem 84 Calculus: Early Transcendentals 7
Suppose that is continuous and , but . Does have a local maximum or minimum at ? Does have a point of inflection at ?
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Chapter 4: Problem 85 Calculus: Early Transcendentals 7
Suppose is differentiable on an interval and for all numbers in except for a single number . Prove that is increasing on the entire interval .
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Chapter 4: Problem 86 Calculus: Early Transcendentals 7
. For what values of is the function increasing on ?
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Chapter 4: Problem 87 Calculus: Early Transcendentals 7
The three cases in the First Derivative Test cover the situations one commonly encounters but do not exhaust all possibilities. Consider the functions whose values at 0 are all 0 and, for (a) Show that 0 is a critical number of all three functions but their derivatives change sign infinitely often on both sides of 0. (b) Show that has neither a local maximum nor a local minimum at 0, has a local minimum, and has a local maximum.
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