Problem 23 (a) Prove that an increasing and... [FREE SOLUTION]

Chapter 12: Problem 23

(a) Prove that an increasing and a decreasing function intersect at most once. (b) Find two continuous increasing functions \(f\) and \(g\) such that \(f(x)=g(x)\) precisely when \(x\) is an integer. (c) Find a continuous increasing function \(f\) and a continuous decreasing function \(g,\) defined on \(R,\) which do not intersect at all.

Short Answer

Expert verified

An increasing and a decreasing function intersect at most once. Functions \(f(x) = \frac{1}{2}x\) and \(g(x) = \frac{1}{2}x + 0.5\) intersect at integer points. Functions \(f(x) = x\) and \(g(x) = -x - 2\) do not intersect.

Step by step solution

Title - proving intersection at most once

Given two functions: an increasing function \(f(x)\) and a decreasing function \(g(x)\). Prove they can intersect at most once. Assume that they intersect at some point \(a\), thus \(f(a) = g(a)\). If they were to intersect again at some point \(b > a\), then because \(f(x)\) is increasing and \(g(x)\) is decreasing, \(f(b) > f(a)\) and \(g(b) < g(a)\). Therefore, \(f(b)\) cannot be equal to \(g(b)\), which contradicts our assumption. Thus they cannot intersect more than once.

Title - finding continuous increasing functions intersecting at integer points

Consider two continuous increasing functions: \(f(x) = \frac{1}{2}x\) and \(g(x) = \frac{1}{2}x + 0.5\). For these functions, \(f(x) = g(x)\) if and only if \(\frac{1}{2}x = \frac{1}{2}x + 0.5 - 0.5\), which simplifies to \(0.5 \times x = 0.5 \times x\). Thus, \(x\) must be an integer.

Title - finding continuous increasing and decreasing functions that do not intersect

Consider the following functions defined on \(R\): \(f(x) = x\) and \(g(x) = -x - 2\). Function \(f(x)\) is continuous and increasing, while \(g(x)\) is continuous and decreasing. For any \(x\) on the real line, \(f(x) = x\) does not intersect with \(g(x) = -x - 2\) because \(x\) will never equal \(-x - 2\). Therefore, these two functions do not intersect at all.

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Key Concepts

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

Increasing Function

An increasing function is a type of function where, as the input value (usually denoted as x) increases, the output value (usually denoted as f(x)) also increases. Think of it like climbing a hill; as you step forward, you keep going higher. Formally, a function f(x) is called increasing if for every two numbers a and b such that a < b, we have f(a) 鈮� f(b).

Here are a few key points to understand increasing functions:

**Strictly Increasing**: If f(a) < f(b) for all a < b, the function is strictly increasing.
**Real-world Examples**: The temperature over the day during summer, the balance of an interest-earning savings account, or the population growth over time.
**Graph**: On a graph, an increasing function slopes upwards as you move from left to right.

In our exercise, one example of an increasing function described is f(x) = x. For any two different points, as x values increase, the function values also increase.

Decreasing Function

A decreasing function is the opposite of an increasing function. Here, as the input value increases, the output value decreases. Imagine walking down a slope; as you continue walking forward, you descend lower. Formally, a function g(x) is called decreasing if for every two numbers a and b such that a < b, we have g(a) 鈮� g(b).

Important points to note about decreasing functions include:

**Strictly Decreasing**: If g(a) > g(b) for all a < b, the function is strictly decreasing.
**Real-world Examples**: The amount of remaining fuel in a tank as you drive, depreciation of a car's value, or the countdown of a timer.
**Graph**: On a graph, a decreasing function slopes downwards as you move from left to right.

In the exercise, an example of a decreasing function is g(x) = -x - 2. As the x values increase, the function values decrease.

Continuous Functions

A continuous function is one where there are no breaks, jumps, or gaps in the function's graph. You can draw the graph of a continuous function without lifting your pen off the paper. Mathematically, a function f(x) is continuous at a point x = c if the following three conditions are satisfied:

**The function is defined at c**: f(c) exists.
**The limit as x approaches c exists**: \(\text{lim}_{x \to c} f(x)\) exists.
**The limit equals the function value**: \(\text{lim}_{x \to c} f(x) = f(c)\).

Some facts about continuous functions:

**No Breaks**: There should be no interruptions in the graph of the function.
**Smooth Transitions**: The function's graph should smoothly transition without any abrupt changes.
**Real-world Examples**: Temperature changes throughout the day, height growth over time, or water flowing from a tap.

In the exercise, both f(x) = \(\frac{1}{2} x\) and g(x) = \(\frac{1}{2} x + 0.5\) are continuous and increasing. Another example provided is f(x) = x, which is continuous and increasing, and g(x) = -x - 2, which is continuous and decreasing. Understanding these properties helps to analyze their intersection points.

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Short Answer

Step by step solution

Title - proving intersection at most once

Title - finding continuous increasing functions intersecting at integer points

Title - finding continuous increasing and decreasing functions that do not intersect

Key Concepts

Increasing Function

Decreasing Function

Continuous Functions

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