Find two numbers whose difference is 76 and whose product is a minimum. (smaller number)
(larger number)
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Step 1 ：Let's denote the two numbers as \(x\) and \(y\).

Step 2 ：We know that the difference between the two numbers is 76, so we can write the equation \(y = x + 76\).

Step 3 ：The product of the two numbers is given by the function \(f(x,y) = xy\). Substituting \(y\) into the function, we get \(f(x) = x(x + 76)\).

Step 4 ：We want to find the minimum of this function. To do this, we can take the derivative of the function, set it equal to zero, and solve for \(x\). This will give us the \(x\)-value at which the function reaches its minimum.

Step 5 ：The derivative of the function \(f(x) = x(x + 76)\) is \(f'(x) = 2x + 76\).

Step 6 ：Setting the derivative equal to zero gives us the critical points: \(2x + 76 = 0\), which simplifies to \(x = -38\).

Step 7 ：Substituting \(x = -38\) back into the equation \(y = x + 76\) gives us \(y = 38\).

Step 8 ：The minimum product is achieved when \(x = -38\) and \(y = 38\). However, since we are looking for two numbers whose difference is 76, we should take the absolute value of \(x\) to get 38. So the two numbers are 38 and -38 + 76 = 38.

Step 9 ：Final Answer: The two numbers are \(\boxed{38}\) and \(\boxed{-38}\).