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Chapter 4: Problem 24

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. \begin{equation} y=x-\sin x, \quad 0 \leq x \leq 2 \pi \end{equation}

Short Answer

Expert verified
Absolute minimum at \((0, 0)\), maximum at \((2\pi, 2\pi)\); inflection point at \((\pi, \pi)\).

Step by step solution

01

Find the Derivative

To find extreme points, we need the first derivative of the function. Differentiate \( y = x - \sin x \) with respect to \( x \):\[\frac{dy}{dx} = 1 - \cos x\]
02

Identify Critical Points

Set the first derivative equal to zero to find critical points:\[1 - \cos x = 0\]Solve for \( x \):\[\cos x = 1\]The solutions within the interval \([0, 2\pi]\) are \( x = 0 \) and \( x = 2\pi \).
03

Second Derivative Test

Compute the second derivative to determine concavity and inflection points. Differentiate the first derivative:\[\frac{d^2y}{dx^2} = \sin x\]Evaluate the second derivative at critical points:- At \( x = 0 \): \( \sin 0 = 0 \)- At \( x = 2\pi \): \( \sin 2\pi = 0 \)Neither point changes concavity, so these are not local extrema. Check the function's endpoints to find points of interest.
04

Check Endpoints for Extrema

Evaluate the function at the endpoints to determine absolute extrema:- At \( x = 0 \): \[y(0) = 0 - \sin 0 = 0\]- At \( x = 2\pi \): \[y(2\pi) = 2\pi - \sin 2\pi = 2\pi\]Since the function is increasing from \( x = 0 \) to \( x = 2\pi \), \( y(0) = 0 \) is the absolute minimum and \( y(2\pi) = 2\pi \) is the absolute maximum.
05

Identify Inflection Points

An inflection point occurs where the second derivative changes sign. Set \( \sin x = 0 \):\[\sin x = 0 \\Rightarrow x = 0, \pi, 2\pi\]Check sign changes of \( \sin x \) around these points:- From \( 0 \leq x < \pi \), \( sin x > 0 \), concave up.- From \( \pi < x \leq 2\pi \), \( sin x < 0 \), concave down.The point \( x = \pi \) is an inflection point because the concavity changes.
06

Graph the Function

To sketch the graph, note the following:- At \( x = 0 \) and \( x = 2\pi \), the function reaches its absolute extremes of 0 and \(2\pi\) respectively.- The function is increasing for \([0, 2\pi]\).- An inflection point at \( x = \pi \) changes concavity from up to down, making the graph flatter near \(x = \pi\).- Plot these points and sketch the curve.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivatives
When tackling mathematical functions, finding the derivative is a crucial step. A derivative represents the rate of change of a function with respect to its variable. For the function given, we first obtained its first derivative, which is expressed as \( \frac{dy}{dx} = 1 - \cos x \). This derivative helps us pinpoint where the slope of the tangent to the function is zero. These are potentially where the function has extreme points, like minima or maxima.
It is particularly important because it tells us whether a function is increasing or decreasing in certain intervals. This information provides insight into the behavior of the function within a specified domain.
Concavity
Concavity refers to the direction in which a graph curves. To determine concavity, I'll use the second derivative, \( \frac{d^2y}{dx^2} = \sin x \).
If this second derivative is positive over an interval, the function is concave up on that interval, meaning the graph holds a cup shape and looks like a smile. However, if it is negative, the function is concave down, resembling a frown. There are no great surprises here, but the information helps shape our understanding of how the graph will look. Knowing the concavity clarifies how the slope is changing and whether an area of the graph is rising or falling at a faster or slower rate.
Inflection Points
An inflection point is where the graph of the function changes its concavity. For our function, this occurs where \( \sin x = 0 \).
In our example, \( x = \pi \) marks one such point since the concavity switches from up to down or vice versa around this point. This change occurs because the sign of \( \sin x \) shifts at \( x = \pi \). Unlike extrema, inflection points do not correspond to a maximum or minimum value of a function. Still, they highlight where the curvature nature changes in the graphical representation, which could influence how you might suggest alterations or predictions regarding the function's behavior.
Critical Points
Critical points are found by setting the first derivative to zero, providing the values where the function's slope might be zero. These values are potentially where the function reaches a peak or a trough.
For our function, by setting \( 1 - \cos x = 0 \), we find that at \( x = 0 \) and \( x = 2\pi \), these critical points offer no local extrema, as confirmed through further testing such as using the second derivative test. In some functions, critical points highlight significant changes in the direction (like peaks and troughs), but here they show other important shifts, such as the endpoints of our interval. By testing relevant intervals and endpoints, we're able to identify the absolute extremas by checking the values of the function at these points.

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Most popular questions from this chapter

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Suppose the derivative of the function \(y=f(x)\) is \begin{equation}y^{\prime}=(x-1)^{2}(x-2)(x-4).\end{equation} At what points, if any, does the graph of \(f\) have a local minimum, local maximum, or point of inflection?

Distance between two ships At noon, ship \(A\) was 12 nautical miles due north of ship \(B .\) Ship \(A\) was sailing south at 12 knots (nautical miles per hour; a nautical mile is 2000 yd) and continued to do so all day. Ship \(B\) was sailing east at 8 knots and continued to do so all day. \begin{equation} \begin{array}{l}{\text { a. Start counting time with } t=0 \text { at noon and express the distance }} \\ \quad{\text { } s \text { between the ships as a function of } t \text { . }} \\ {\text { b. How rapidly was the distance between the ships changing at }} \\ \quad{\text { noon? One hour later? }} \\\ {\text { c. The visibility that day was } 5 \text { nautical miles. Did the ships }} \\ \quad{\text { ever sight each other? }} \\ {\text { d. Graph } s \text { and } d s / d t \text { together as functions of } t \text { for }-1 \leq t \leq 3} \\ \quad {\text { using different colors if possible. Compare the graphs and }} \\ \quad {\text { reconcile what you see with your answers in parts (b) and (c). }} \\ {\text { e. The graph of } d s / d t \text { looks as if it might have a horizontal asymptote }} \\ \quad {\text { in the first quadrant. This in turn suggests that } d s / d t} \\ \quad {\text { approaches a limiting value as } t \rightarrow \infty . \text { What is this value? }} \\ \quad {\text { What is its relation to the ships' individual speeds? }} \end{array} \end{equation}
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