/*! 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 3 Let \(f(x)=\sum_{k=1}^{n}|x-k| .... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 3

Let \(f(x)=\sum_{k=1}^{n}|x-k| .\) Find all absolute minimizers of the function.

Short Answer

Expert verified
The absolute minimizers are the median(s) of \( \{1, 2, ..., n\} \).

Step by step solution

01

Understanding the Function

The given function is the sum of absolute values: \( f(x) = \sum_{k=1}^{n} |x-k| \). This means that for any position \( x \), the function \( f(x) \) measures the total distance from \( x \) to each integer from 1 to \( n \).
02

Visualize with Small Values of n

Let's consider a small example, like \( n = 3 \). Here, \( f(x) = |x-1| + |x-2| + |x-3| \). For \( x = 1 \), \( f(1) = 0 + 1 + 2 = 3 \); for \( x = 2 \), \( f(2) = 1 + 0 + 1 = 2 \); and for \( x = 3 \), \( f(3) = 2 + 1 + 0 = 3 \). The minimized value appears at \( x = 2 \).
03

Test Patterns for More General n

The function \( f(x) \) will always form a V pattern, with the lowest point at the median of the set \( \{1, 2, ..., n\} \). This is because in a sequence of numbers, the median minimizes the sum of absolute deviations.
04

Find the Median

If \( n \) is an odd number, the median is the central number, \( x = \frac{n+1}{2} \). If \( n \) is even, both central numbers \( x = \frac{n}{2} \) and \( x = \frac{n}{2} + 1 \) serve as minimizers because the sum is symmetrically balanced between these points.
05

Conclusion Based on Calculations

The absolute minimizers are the median(s) of the set of integers \( \{1, 2, ..., n\} \). For odd \( n \), it's a single value, \( x = \frac{n+1}{2} \). For even \( n \), two values minimize \( f(x) \), namely \( x = \frac{n}{2} \) and \( x = \frac{n}{2} + 1 \).

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.

Absolute Values
Absolute value is a fundamental concept in mathematics. It refers to the distance of a number from zero on the number line, without considering direction. Essentially, it is a measure of magnitude irrespective of sign. For any real number \(a\), the absolute value is denoted \(|a|\). This is defined as \(|a| = a\) if \(a \geq 0\) and \(|a| = -a\) if \(a < 0\).
  • Absolute values help in ensuring non-negative outcomes when measuring distances, which is crucial in distance measurement problems.
  • They're instrumental in fields such as engineering and physics where distance and magnitude computations are essential.
When working with absolute values, always consider that they effectively "erase" the direction, meaning both \(10\) and \(-10\) have an absolute value of 10.
Median
The median is a statistical measure that identifies the middle value of a data set when arranged in order. It's different from the mean, which calculates an average. The median provides a better measure of central tendency in skewed distributions or when outliers are present.
  • To find the median, sort the numbers in ascending order and identify the middle value.
  • For an odd number of observations, the median is the middle number.
  • For an even number, it is the average of the two middle numbers.
When applied to the function \(f(x) = \sum_{k=1}^{n} |x-k|\), the median is key in identifying the point that minimizes the function, because it minimizes the sum of absolute deviations.
Summation
Summation is the process of adding a sequence of numbers. It's a basic operation in algebra denoted by the symbol \(\Sigma\), representing the aggregate sum.
  • This concept is essential in calculus, statistics, and series representation.
  • It helps in functions like \(f(x) = \sum_{k=1}^{n} |x-k|\), which assesses the total distance from a point \(x\) to each of the integers from 1 to \(n\).
Summation allows for simplifying complex problems by breaking them down into manageable parts, like finding total deviations in a data set.
Distance Measurement
Distance measurement in mathematics involves calculating how far apart two points are. Absolute values are widely used for this purpose since they provide straightforward non-negative results.
  • It is foundational in both mathematical theory and practical fields such as physics and engineering.
  • The function \(f(x) = \sum_{k=1}^{n} |x-k|\) is a classic example of utilizing distance measurement, quantifying how far a point \(x\) is from each element in a sequence.
Understanding distance measurement principles is essential for accurately interpreting the function \(f(x)\), helping to determine where the distance (or sum of distances) is minimized.

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

Determine if each of the following functions is differentiable at \(0 .\) Justify your answer. (a) \(f(x)=\left\\{\begin{array}{ll}x^{2}, & \text { if } x \in \mathbb{Q} \\\ x^{3}, & \text { if } x \notin \mathbb{Q}\end{array}\right.\) (b) \(f(x)=[x] \sin ^{2}(\pi x)\) (c) \(f(x)=\cos (\sqrt{|x|})\). (d) \(f(x)=x|x|\).

Let \(n\) be a fixed positive integer. (a) Suppose \(a_{1}, a_{2}, \ldots, a_{n}\) satisfy $$ a_{1}+\frac{a_{2}}{2}+\cdots+\frac{a_{n}}{n}=0 $$ Prove that the equation $$ a_{1}+a_{2} x+a_{3} x^{2}+\cdots+a_{n} x^{n-1}=0 $$ has a solution in (0,1) . (b) Suppose \(a_{0}, a_{1}, \ldots, a_{n}\) satisfy $$ \sum_{k=0}^{n} \frac{a_{k}}{2 k+1}=0 $$ Prove that the equation $$ \sum_{k=0}^{n} a_{k} \cos (2 k+1) x=0 $$ has a solution on \(\left(0, \frac{\pi}{2}\right)\).

Consider the function $$ f(x)=\left\\{\begin{array}{ll}x^{2} \sin \frac{1}{x}+c x, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right. $$ where \(00,\) the function \(f^{\prime}\) changes its sign on \((-\alpha, \alpha)\).

Determine the values of \(x\) at which each function is differentiable. (a) \(f(x)=\left\\{\begin{array}{ll}x \sin \frac{1}{x}, & \text { if } x \neq 0 \\\ 0, & \text { if } x=0\end{array}\right.\) (b) \(f(x)=\left\\{\begin{array}{ll}x^{2} \sin \frac{1}{x}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.\)

Prove the following: (a) If \(a, b\) are nonnegative real numbers, then $$ \frac{a+b}{2} \geq \sqrt{a b} $$ (b) If \(a_{1}, a_{2}, \ldots, a_{n},\) where \(n \geq 2,\) are nonnegative real numbers, then $$ \frac{a_{1}+a_{2}+\cdots+a_{n}}{n} \geq\left(a_{1} \cdot a_{2} \cdots a_{n}\right)^{1 / n} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied 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.