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Chapter 1: Problem 42

Give an example of two increasing functions whose product is not increasing. [Hint: There are no such examples where both functions are positive everywhere.]

Short Answer

Expert verified
The two increasing functions are \(f(x) = x\) and \(g(x) = x^3\). Their product \(h(x) = f(x) * g(x) = x^4\) is an increasing function for positive values of \(x\) but is decreasing for negative values of \(x\).

Step by step solution

01

Find the product of the two functions

Multiply the two functions \(f(x)\) and \(g(x)\) as follows: \(h(x) = f(x) * g(x) = x * x^3 = x^4\)
02

Check if the product is an increasing function

To check if \(h(x) = x^4\) is an increasing function, we need to find its first derivative: \(\frac{dh(x)}{dx} = 4x^3\) Now, let's see if the derivative is positive everywhere: For positive values of \(x\), \(\frac{dh(x)}{dx} \gt 0\), indicating that the function is increasing in that interval. For negative values of \(x\), \(\frac{dh(x)}{dx} \lt 0\), which means the function is decreasing in that interval. So, we have successfully found two increasing functions, \(f(x) = x\) and \(g(x) = x^3\), whose product is not increasing everywhere, as \(h(x) = x^4\) is decreasing for negative values of \(x\).

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