Question

Find two positive numbers satisfying the given requirements. The product is 192 and the sum is a minimum.

Short answer

The two positive numbers satisfying the given conditions are \(x = 8\sqrt{3}\) and \(y = 8/\sqrt{3}\).

Step-by-step solution

Formulate the Equations

Define the numbers as \(x\) and \(y\). The product condition gives us the equation \(xy = 192\). From this equation, we can express \(y\) as \(y = 192/x\).

Define the Optimization Function

We need to minimize the sum of \(x\) and \(y\), hence the sum \(s = x + y\) can be expressed as \(s = x + 192/x\) using the equation from Step 1.

Differentiate and Find Critical Points

Differentiate \(s\) with respect to \(x\) to get \(ds/dx\). Find values of \(x\) for which \(ds/dx = 0\) or \(ds/dx\) is undefined. These values correspond to the critical points.

Evaluate the Optimization Function

Plug the critical points obtained from Step 3 into the original optimization function to find the value of \(y\) and to ensure that the minimum sum is obtained.

Check the Values

Ensure that the values of \(x\) and \(y\) satisfy the original conditions - they must be positive and their product should be 192.