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Chapter 3: Problem 63

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 minimum product of a pair of numbers whose difference is 16 is -64.

Step by step solution

01

Derive the given equation from the problem

Create a function from the given problem. Let's denote one number as \(x\) and the other number as \(x+16\). The product of these two should be as small as possible. Hence, the function \(f(x) = x(x + 16)\) is to be minimized.
02

Simplification of Equation

Simplify the function equation and it becomes, \(f(x) = x^2 + 16x\).
03

Calculate the Derivative

For finding the minimum value, differentiate the equation \(f'(x) = 2x + 16\).
04

Set the derivative equal to zero and solve for \(x\)

To find the minimum value, set the derivative to zero and solve for \(x\). Afterwards, \(2x + 16 = 0\) gives \(x = -8\).
05

Verify that it gives minimum value via second derivative test

Take the second derivative of the function: \(f''(x) = 2\), which is always positive for all real \(x\). This means that the function \(f(x)\) is concave up for all \(x\), and thus, \(x = -8\) indeed minimizes \(f(x)\).
06

Calculate minimum product value

Substitute \(x = -8\) into the original function to get the minimum product. Hence, \(f(-8) = (-8)*(-8 + 16) = -8 * 8 = -64.\) Hence, the minimum product is -64.

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