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Chapter 11: Problem 62

Among all pairs of numbers whose difference is \(24,\) find a pair whose product is as small as possible. What is the minimum product?

Short Answer

Expert verified
The minimum product is 0. This occurs when the two numbers are 0 and 24.

Step by step solution

01

Set Up Equations

Let the pair of numbers be \(x\) and \(y\). The conditions that their difference is 24 and that the product is to be minimized can be expressed as two equations: 1) \(y - x = 24\) and 2) \(P = xy\) where \(P\) is the product that we want to minimize.
02

Express P in Terms of One Variable

Using the first equation, we can express \(y\) as \(y = x + 24\). Substitute this into the second equation to get the expression for \(P\) in terms of \(x\) only: \(P = x(x + 24)\).
03

Calculate Derivative and Find Critical Points

Take the derivative of \(P\) with respect to \(x\), set it to zero and solve for \(x\). The derivative \(P'\) is \(2x + 24\). Setting \(P' = 0\) gives \(x = -12\). However, \(x\) can't be negative as it's the difference between two numbers, so this solution is discarded.
04

Investigate the Extrema

In this case, the potential extrema will happen at the boundaries of the possible values for \(x\), that is, \(x = 0\) and \(x = 24\). Evaluate \(P\) at these boundary points: \(P(0) = 0 \times 24 = 0\) and \(P(24) = 24 \times 48 = 1152\).
05

Identify the Minimum

Since we are asked to find the pair of numbers with the smallest product, the minimum is the smallest value of \(P\). Thus, the minimum product is when \(P = 0\).

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