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Chapter 11: Problem 61

Among all pairs of numbers whose difference is \(16,\) find a pair whose product is as small as possible. What is the minimum product?

Short Answer

Expert verified
The smallest possible product is -64 when the pair is (8, -8).

Step by step solution

01

Represent the problem in terms of equations

Let's denote one of the numbers as \(x\). It is said that the difference between the two numbers is 16. If one number is \(x\), the other number would be \(x - 16\). Now, we seek to minimize the product, which is \(x(x - 16) = x^2 - 16x\). This is our function that we want to minimize.
02

Derive the function

To find the minimum of this function, we can use calculus. The first step is to find the derivative of the function. The derivative of \(x^2 - 16x\) is \(2x - 16\).
03

Find the zero of the derivative

A minimum or maximum of a function corresponds to the points where the derivative is zero. Thus, set the derivative equal to zero and solve for \(x\). \(2x - 16 = 0\) Solving this gives us \(x=8\). Hence, our pair is \(8, -8\). Note: We understand that it's at minimum due to the positive value of the second derivative \(2 > 0\), which indicates a local minima.
04

Find the minimum product

Plug these numbers back into our product equation: The minimum product is thus \(8*(-8) = -64\).

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