/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 2 Classify each property of \(f(x)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 2

Classify each property of \(f(x)\) as pointwise, local, global, or nonexistent. (a) even function (b) has a supremum on \([a, b]\) (c) locally odd function (d) \(y=f(x)\) satisfies the equation \(y^{5}+2 x y-3 x=0\) on \(I\) (e) is a linear function on \(I\) (f) has a maximum on \([a, b]\) (g) \(f(x) \neq 3\) for \(x \in I\) (h) has a maximum on \(I\) at the point \(a\) (i) \(\max _{I} f(x)=5\) (j) \(f(x)\) has a zero on \(I\)

Short Answer

Expert verified
(a) Global, (b) Global, (c) Local, (d) Pointwise, (e) Global, (f) Global, (g) Global, (h) Pointwise, (i) Global, (j) Pointwise.

Step by step solution

01

Classify even function (a)

An even function is symmetric about the y-axis, meaning for every \( x \), \( f(x) = f(-x) \). This property holds for the entire domain of \( f \). Thus, evenness is a global property.
02

Classify has a supremum on [a, b] (b)

A supremum on an interval \( [a, b] \) refers to the least upper bound of the function values in that interval. Since it concerns every point in \( [a, b] \), it is a global property.
03

Classify locally odd function (c)

A function is locally odd at a point \( c \) if there exists a neighborhood around \( c \) where \( f(x) = -f(-x) \). This property pertains to behavior around specific points, making it a local property.
04

Classify equation \\( y^{5}+2xy-3x=0 \\) (d)

The presence of an equation that \( y = f(x) \) must satisfy does not describe symmetry or extrema properties but rather a pointwise condition applying to each \( x \) individually. This is a pointwise property.
05

Classify is a linear function on I (e)

A linear function over an interval \( I \) has the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants. This characterizes the entire interval \( I \) and not specific points or segments, making it a global property.
06

Classify has a maximum on [a, b] (f)

Having a maximum on \( [a, b] \) means there's a point in this interval where \( f \) reaches its highest value. As this concerns all points in \( [a, b] \), it is a global property.
07

Classify \\( f(x) \neq 3 \\) for \\( x \in I \\) (g)

The property that \( f(x) eq 3 \) for all \( x \in I \) applies to every point within \( I \), making it a global property.
08

Classify has a maximum on I at the point a (h)

Having a maximum at a single point \( a \) in the interval \( I \) does not describe behavior on the entire interval, but rather at a specific point. Thus, it is a pointwise property.
09

Classify \\( \max_{I} f(x)=5 \\) (i)

The statement that the maximum value of \( f \) over the interval \( I \) is 5 is a global property because it concerns the function's behavior over the entire interval.
10

Classify \\( f(x) \\) has a zero on \\( I \\) (j)

If \( f(x) \) has a zero on \( I \), it means there exists some point in the interval where \( f(x) = 0 \). This property applies at specific points in \( I \), making it a pointwise property.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Pointwise Properties
Pointwise properties apply to individual points in the function's domain. They are concerned with very specific behaviors at specific values rather than on intervals. For instance, if a pointwise property holds at point \( a \), it may not necessarily hold at any other point.
  • For example, the equation \( y^{5} + 2xy - 3x = 0 \) describes a pointwise property, as it's a condition that individual points \( (x, y) \) must satisfy.
  • Another instance is having a maximum at a single point \( a \) in an interval, which also demonstrates pointwise behavior, as it focuses on one specific location, rather than the entirety of the interval.
  • Similarly, if \( f(x) \) has a zero on \( I \), this indicates there exists at least one specific point in \( I \) where \( f(x) = 0 \).
These pointwise properties tell us about specific, isolated behaviors of the function, rather than general trends or patterns.
Local Properties
Local properties describe behaviors of functions around a specific point, within a neighborhood. They give us insight into how the function behaves within a small scope around a given point.
  • A locally odd function is a classic example: there exists a neighborhood around a point \( c \) where the function behaves such that \( f(x) = -f(-x) \). It doesn't apply broadly; only near point \( c \).
Local properties are all about proximity, focusing on how the function fares in small areas around specific points, rather than worrying about its behavior across an entire domain or interval.
Global Properties
Global properties pertain to behaviors that span the entire domain or a specified interval of a function. They provide information about how the function behaves overall, rather than focusing on specific points.
  • For example, evenness (i.e., the function \( f(x) = f(-x) \)) is a global property that describes the entire function's symmetry about the y-axis.
  • The supremum on an interval \([a, b]\) addresses the least upper boundary across that entire interval, marking it as a global concern.
  • Having a maximum on \([a, b]\) is also a global trait, indicating where the function reaches its highest value within that interval.
  • A linear function over an interval, defined as \( f(x) = mx + b \), portrays a global property because it consistently describes the function throughout the interval.
  • When a property claims that \( \max_{I} f(x) = 5 \), it's describing the maximum value the function attains on the interval \( I \), making it a global characteristic.
  • Similar to the aforementioned examples, \( f(x) eq 3 \) for every \( x \in I \) states a consistent attribute across the whole interval, indicating it's a global property.
Global properties assist us in grasping a comprehensive understanding of the function's overall behavior over its domain or some specified section.
Supremum and Maximum
The concepts of supremum and maximum are often discussed together, yet they describe different characteristics of function behavior on intervals. **Supremum**:
  • The supremum (least upper bound) of a set in an interval is the smallest number that is greater than or equal to every number in that set. It does not need to belong to the set.
  • If a function has a supremum on an interval \([a, b]\), it refers to this least upper boundary that caps all function values within that interval.
**Maximum**:
  • The maximum of a function on an interval is the greatest function value actually achieved over that interval.
  • If a function has a maximum on \([a, b]\) or at a point on interval \( I \), it means there is a specific point within the interval where the function reaches this peak value.
  • For example, \( \max_{I} f(x) = 5 \) would indicate that 5 is the highest value \( f \) attains over \( I \) and it does indeed achieve this highest value.
Understanding these traits can help define the boundaries and peaks of function behavior on a particular span or domain.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For each, tell if it is bounded (on its domain); if so, give bounds. $$\text { (a) } x \sin x$$ (b) \(\frac{x^{2}+1}{x^{2}+4}\) (c) \(\frac{x-2}{x+1}\)

Find the interval on which this second-order approximation is valid: $$(1-x)\left(1+x^{2}\right) \approx 1-x+x^{2}, \quad \epsilon=.001$$

Tell whether each of the following is true or false (no proofs required): (a) \(\sin x \leq x\) for \(x \approx 0\) (b) \(e^{-x}<.01\) for \(x \gg 1\); (c) if \(f(x)\) is a polynomial, then \(1 / f(x)\) is defined at \(\infty\) and \(-\infty\); (d) given \(\epsilon>0, \quad \sin x \approx x\) for \(x \approx 0\); (e) \(x^{2}\) is locally bounded at any \(x_{0}\), but not locally bounded at \(\infty\); (f) \(x^{2}\) is locally monotone at any \(x_{0}\), and also at \(\infty\) and \(-\infty\).

If \(f(x) \approx 1+x\) and \(g(x) \underset{\epsilon}{\approx} 1-x\) for \(|x| \leq 1\), estimate $$\int_{-1}^{1}(f(x)+g(x)) d x$$

Give the values of sup, max, inf, and \(\min f(x)\) on \(I\), if they exist: $$f(x)=\frac{x}{1+x} ; \quad I=[0, \infty), \quad(0, \infty), \quad(-1, \infty), \quad(-\infty, \infty)$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.