Minimizing the Product of Two Numbers with a Fixed Difference

Consider the problem of finding two numbers whose difference is 256 and whose product is minimized.

Defining the Problem

Let's denote the two numbers as x and y, with the condition that:

x - y 256

To proceed, we express one variable in terms of the other. For instance:

x y 256

Formulating the Product

Next, we need to minimize the product P of these two numbers:

P x y (y 256) y y^2 256y

This forms a quadratic equation in the standard form:

P y^2 256y

To find the minimum value of this quadratic function, we can use calculus or complete the square. Let's use calculus for this case.

Using Calculus to Find the Minimum

We take the derivative of P with respect to y and set it equal to zero to find the critical points:

P' 2y 256

2y 256 0

y -128

Substituting y -128 back into the equation for x:

x y 256 -128 256 128

Verification

To verify that these values minimize the product:

The difference between the two numbers: 128 - (-128) 256, which is correct. The product of the two numbers: 128 (-128) -16384, which is the minimum value.

Conclusion

Therefore, the two numbers that satisfy the given conditions are 128 and -128 with their product minimized at -16384.

This example demonstrates the process of applying calculus to find the minimum product of two numbers given their difference.