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Magazine Chapter 4: Problem 14
(a) Sketch the graph of a function on f21, 2g that has an absolute maximum but no local maximum. (b) Sketch the graph of a function on f21, 2g that has a local maximum but no absolute maximum.
Short Answer
Expert verified
(a) Use a linear function increasing to 2; (b) Use a parabolic function peaking at a local point.
Step by step solution
01
Understand the Problem
We need to sketch two separate graphs based on the given conditions for the function on the interval \([-1, 2]\). The first sketch (a) requires an absolute maximum but no local maximum, while the second sketch (b) requires a local maximum but no absolute maximum.
02
Sketch Graph for (a)
For part (a), we want a function with an absolute maximum but no local maximum on \([-1, 2]\). This means that the absolute maximum occurs at the endpoint, providing a simple linear function that increases across the interval. Let's use the function \(f(x) = x + 1\). This has an absolute maximum at \(x = 2\) with no other points considered a local maximum.
03
Verify for (a)
Check that the function \(f(x) = x + 1\) on \([-1, 2]\) achieves an absolute maximum only at \(x = 2\) and no other points as local maxima. As it occurs at the endpoint and the slope is constant, there are no local extrema within the interval.
04
Sketch Graph for (b)
For part (b), we need a function with a local maximum but no absolute maximum. This means the local maximum cannot be the largest value in the entire interval. An example is a parabolic function like \(f(x) = -(x+0.5)^2 + 2\), which has a peak inside the interval.
05
Verify for (b)
Examine the function \(f(x) = -(x+0.5)^2 + 2\). It has a local maximum at \(x = -0.5\), with a value of 2. However, as we reach \(x = 2\), the function value never exceeds the local maximum value, ensuring there's no absolute maximum within \([-1, 2]\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Absolute Maximum
An absolute maximum of a function in a given interval is the highest value that the function achieves over that entire range. This means that no other point within that interval has a function value greater than this point. The concept of an absolute maximum is an important aspect of understanding the overall behavior of the function.
In exercise (a), an example of achieving an absolute maximum without a local maximum is the function \( f(x) = x + 1 \) on the interval \([-1, 2]\). Here, the absolute maximum occurs at \( x = 2 \), and the value is \( f(2) = 3 \).
In exercise (a), an example of achieving an absolute maximum without a local maximum is the function \( f(x) = x + 1 \) on the interval \([-1, 2]\). Here, the absolute maximum occurs at \( x = 2 \), and the value is \( f(2) = 3 \).
- The maximum is absolute because, within the interval, no other point has a higher value.
- Since this occurs at the endpoint and not within the openness of the interval, it is not considered a local maximum.
Local Maximum
A local maximum is a point within a function's domain where the function value is greater than at any nearby points. This means that the neighborhood around this point does not exceed it in value, even if it's not the highest in the entire range.
In exercise (b), the function \( f(x) = -(x+0.5)^2 + 2 \) has a local maximum at \( x = -0.5 \). At this point, \( f(-0.5) = 2 \) is greater than any other point in its immediate vicinity, though this maximum is not absolute across the interval \([-1,2]\).
In exercise (b), the function \( f(x) = -(x+0.5)^2 + 2 \) has a local maximum at \( x = -0.5 \). At this point, \( f(-0.5) = 2 \) is greater than any other point in its immediate vicinity, though this maximum is not absolute across the interval \([-1,2]\).
- Local maxima occur within the bounds of the function, not necessarily the end.
- This allows a curve, such as a parabola, to have peaks even though higher values may exist elsewhere.
Linear Function
A linear function represents a straight line and is one of the simplest forms of a function. It has the format \( f(x) = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. This type of function has constant rates of change, and its graph is always a straight line.
In the solution to exercise (a), the function \( f(x) = x + 1 \) is linear within \([-1, 2]\). Here's how it relates:
In the solution to exercise (a), the function \( f(x) = x + 1 \) is linear within \([-1, 2]\). Here's how it relates:
- The function is straightforward and increases steadily across the interval.
- Because the function is linear and increasing, it reaches its maximum at the endpoint \( x = 2 \) with no dips or bumps to create local maxima.
Parabolic Function
A parabolic function is defined by a quadratic equation and its graph forms the shape of a parabola. These functions take the form \( f(x) = ax^2 + bx + c \). Depending on the coefficient \( a \), the parabola can open upwards or downwards:
For exercise (b), the function \( f(x) = -(x+0.5)^2 + 2 \) is a downward opening parabola. Here is why:
- When \( a > 0 \), it opens upwards, resembling a smile.
- When \( a < 0 \), it opens downwards, akin to a frown.
For exercise (b), the function \( f(x) = -(x+0.5)^2 + 2 \) is a downward opening parabola. Here is why:
- The negative sign before \( (x+0.5)^2 \) indicates an inverted parabola.
- It features a peak or local maximum at \( x = -0.5 \) without an absolute maximum across the interval \([-1, 2]\).
