Question

Find two numbers whose difference is 44 and whose product is minimized. Then determine the minimum product.

A. Set up an equation for the difference and the product.
B. Solve the system of equations.
C. Verify the solution by testing for the minimum product.
D. Interpret the result in the context of the problem.

Answer 1

To find two numbers whose difference is 44 and whose product is minimized, assume the first number is x and the second number is x + 44. Take the derivative of the product function and set it equal to 0 to find the x-coordinate of the minimum product. Plugging this value back into the original equation gives us the minimum product.

Step-by-step explanation:

1. To find two numbers whose difference is 44, let's assume the first number is x, and the second number is x + 44.
2. The product of these two numbers is x * (x + 44), which we want to minimize. So, our equation is f(x) = x * (x + 44).
3. To find the minimum product, we can use calculus by taking the derivative of f(x) and setting it equal to 0. The critical point we find will give us the x-coordinate of the minimum product.
4. We solve f'(x) = 0 to find that x = -22. Plugging this value back into f(x), we can find the minimum product.
5. The minimum product is (-22) * (-22 + 44) = -484.
6. Therefore, the two numbers whose difference is 44 and whose product is minimized are -22 and 22, and the minimum product is -484.