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Chapter 1: Problem 43

Give an example of two decreasing functions whose product is increasing.

Short Answer

Expert verified
An example of two decreasing functions whose product is increasing are \(f(x) = -x\) and \(g(x) = -x - 1\). Their product is \(h(x) = f(x) \cdot g(x) = x(x + 1)\), which is an increasing function.

Step by step solution

01

Choose the first decreasing function

Let's consider the linear function \(f(x) = -x\), which is a decreasing function.
02

Choose the second decreasing function

Now, let's consider another linear function \(g(x) = -x - 1\), which is also a decreasing function.
03

Compute the product of the two decreasing functions

Now let's compute the product of the two functions: \[h(x) = f(x) \cdot g(x) = (-x) \cdot (-x - 1).\]
04

Simplify the product and check the sign

Simplify the expression for the product function: \[h(x) = (-x)\cdot(-x - 1) = x(x + 1).\] Since \(x\) and \(x + 1\) are both negative for negative values of \(x\), their product, \(h(x)\), will be positive. Additionally, as both \(x\) and \(x + 1\) become less negative as \(x\) increases, the product will also increase. Thus, the functions \(f(x) = -x\) and \(g(x) = -x - 1\) are decreasing functions whose product, \(h(x) = x(x + 1)\), is an increasing function.

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