Question: A cryptography expert is deciphering a computer code. To do this, the expert needs to minimize the…

A cryptography expert is deciphering a computer code. To do this, the expert needs to minimize the product of a positive rational number and a negative rational number, given that the positive number is exactly 5 greater than the negative number. What final product is the expert looking for?

Solution

The problem involves minimizing the product of a positive rational number and a negative rational number, where the positive number is exactly 5 greater than the negative number. Let the negative rational number be \( x \). Then the positive rational number is \( x + 5 \). The product \( P \) is given by: \[ P = x(x + 5) \] Simplifying, we have: \[ P = x^2 + 5x \] To find the minimum product, take the derivative of \( P \) with respect to \( x \): \[ \frac{dP}{dx} = 2x + 5 \] Set the derivative equal to zero to find the critical points: \[ 2x + 5 = 0 \] Solve for \( x \): \[ 2x = -5 \] \[ x = -\frac{5}{2} \] Substitute back to find the product: The positive number is \( x + 5 = -\frac{5}{2} + 5 = \frac{5}{2} \). Then the product \( P \) is: \[ P = -\frac{5}{2} \times \frac{5}{2} = -\frac{25}{4} \] Thus, the minimized product is \(-\frac{25}{4}\).