Problem 64 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The minimum product is -144.

Step 1: Express \(y\) in terms of \(x\)

Since the difference between the pair of numbers is 24, express \(y\) as \(y = x - 24\), where \(x\) and \(y\) are the two numbers.

Step 2: Form the function \(f(x)\) that gives the product \(xy\)

The product of the two numbers is given by the function \(f(x) = x * (x - 24)\). This simplifies to \(f(x) = x^2 - 24x\).

Step 3: Find the minimum of \(f(x)\)

Since the coefficient of \(x^2\) is positive, \(f(x)\) will have a minimum at the point where the derivative \(f'(x)\) equals zero. Differentiate \(f(x)\) to get \(f'(x) = 2x - 24\), and solve for \(x\) by setting \(f'(x) = 0\). This gives \(x = 12\).

Step 4: Calculate the minimum product

Substitute \(x = 12\) into the function \(f(x)\) to get the minimum product. This gives \(f(12) = 12^2 - 24*12 = -144\).