Find two positive numbers x & y such that `x + y = 60` and xy is maximum.

Answer

To solve the problem of finding two positive numbers \( x \) and \( y \) such that \( x + y = 60 \) and the product \( xy \) is maximized, we can follow these steps:

### Step 1: Set up the equations
We start with the equation given in the problem:
\[ x + y = 60 \]
From this equation, we can express \( y \) in terms of \( x \):
\[ y = 60 - x \]

### Step 2: Write the product function
Next, we want to maximize the product \( xy \). Substituting \( y \) from the previous step, we get:
\[ P = xy = x(60 - x) = 60x - x^2 \]
This is a quadratic function in terms of \( x \).

### Step 3: Differentiate the product function
To find the maximum value of \( P \), we need to differentiate it with respect to \( x \):
\[ \frac{dP}{dx} = 60 - 2x \]

### Step 4: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:
\[ 60 - 2x = 0 \]
Solving for \( x \):
\[ 2x = 60 \implies x = 30 \]

### Step 5: Find the corresponding value of \( y \)
Now that we have \( x \), we can find \( y \):
\[ y = 60 - x = 60 - 30 = 30 \]

### Step 6: Verify if it is a maximum
To confirm that this point gives a maximum, we can check the second derivative:
\[ \frac{d^2P}{dx^2} = -2 \]
Since the second derivative is negative, this indicates that the function \( P \) has a maximum at \( x = 30 \).

### Conclusion
Thus, the two positive numbers \( x \) and \( y \) that maximize the product \( xy \) while satisfying \( x + y = 60 \) are:
\[ x = 30, \quad y = 30 \]