Problem 63 - College Algebra

Question

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

Step-by-Step Solution

Answer

The minimum product is -64, achieved by the pair of numbers 8 and -8.

Step 1: Formulation of the problem

Let's denote the two numbers as x and y. According to the problem, \(x - y = 16\). Let's express y in terms of x from this equation: \(y = x - 16\). The product of these two numbers can be expressed as a function of x: \(P(x) = x * (x - 16)\). Our goal is to find the minimum of the function P.

Step 2: Find the derivative of the function

To find the minimum of the function P, we can use the method of setting first derivative equal to zero and solving for x. The derivative of the function \(P(x) = x^2 -16x\) is computed as \(P'(x) = 2x - 16\). Set the derivative equal to zero and solve for x: \(2x - 16 = 0\), which gives \(x = 8\). This is the number that minimizes the product when the difference between the two numbers is 16.

Step 3: Verify the solution

We have found that x=8 minimizes the product. It's important to verify that this value indeed gives a minimum. We do this by examining the value of the second derivative at x=8. The second derivative of the function \(P(x) = x^2 - 16x\) is \(P''(x)=2\), which is always positive, confirming that x=8 gives a minimum. The corresponding value for y using \(y = x - 16\) is \(y = 8 - 16 = -8\). Thus, the pair of numbers that minimize the product when their difference is 16 are 8 and -8.

Step 4: Compute the minimum product

Now we find the minimum product by substituting x = 8 and y = -8 into the equation \(P = xy\). This gives \(P_min = 8 * -8 = -64\).