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Chapter 1: Problem 41

Show that the sum of two increasing functions is increasing.

Short Answer

Expert verified
Given two increasing functions f(x) and g(x), we want to show that their sum h(x) = f(x) + g(x) is also increasing. Based on the definition of increasing functions, for any x1 < x2, we have f(x1) < f(x2) and g(x1) < g(x2). Analyzing h(x) at points x1 and x2, we get h(x1) = f(x1) + g(x1) and h(x2) = f(x2) + g(x2). Adding the inequalities, we obtain h(x1) < h(x2) for any x1 < x2, proving that h(x) is indeed an increasing function.

Step by step solution

01

Define the functions and their properties

Let f(x) and g(x) be two increasing functions. By the definition of increasing functions, for any x1 < x2, we have: 1. f(x1) < f(x2) 2. g(x1) < g(x2) Now, let h(x) = f(x) + g(x) be the sum of these two increasing functions. We need to prove that h(x) is increasing, which means for any x1 < x2, h(x1) < h(x2).
02

Analyze the sum function h(x)

Let's analyze the function h(x) at the points x1 and x2 where x1 < x2. Since h(x) is the sum of f(x) and g(x), we can write h(x1) and h(x2) as: 1. h(x1) = f(x1) + g(x1) 2. h(x2) = f(x2) + g(x2) Now, to prove that h(x) is increasing, we need to show that h(x1) < h(x2).
03

Apply the properties of increasing functions

We know that both f(x) and g(x) are increasing functions; therefore, f(x1) < f(x2) and g(x1) < g(x2). Let's add these two inequalities: f(x1) + g(x1) < f(x2) + g(x2) Notice that the left-hand side is equal to h(x1) and the right-hand side is equal to h(x2). Thus, we get: h(x1) < h(x2)
04

Conclude the proof

We have shown that for any x1 < x2, h(x1) < h(x2). This means that h(x) is an increasing function. Therefore, we have proven that the sum of two increasing functions is also an increasing function.

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