Find the two numbers with maximum product and whose sum is 24.

Answer

To find the two numbers with maximum product whose sum is 24, we can follow these steps:

### Step 1: Define the Variables
Let the two numbers be \( x \) and \( y \). According to the problem, we have:
\[ x + y = 24 \]

### Step 2: Express One Variable in Terms of the Other
From the equation \( x + y = 24 \), we can express \( y \) in terms of \( x \):
\[ y = 24 - x \]

### Step 3: Write the Product Function
The product \( P \) of the two numbers \( x \) and \( y \) can be expressed as:
\[ P = x \cdot y = x(24 - x) = 24x - x^2 \]

### Step 4: Differentiate the Product Function
To find the maximum product, we need to differentiate \( P \) with respect to \( x \):
\[ \frac{dP}{dx} = 24 - 2x \]

### Step 5: Set the Derivative to Zero
To find the critical points, set the derivative equal to zero:
\[ 24 - 2x = 0 \]
Solving for \( x \):
\[ 2x = 24 \implies x = 12 \]

### Step 6: Find the Corresponding Value of \( y \)
Using the value of \( x \) in the equation for \( y \):
\[ y = 24 - x = 24 - 12 = 12 \]

### Step 7: Conclusion
The two numbers that give the maximum product are:
\[ \text{The numbers are } 12 \text{ and } 12. \]

### Step 8: Verify Maximum Product
To confirm that this is a maximum, we can check the second derivative:
\[ \frac{d^2P}{dx^2} = -2 \]
Since \(\frac{d^2P}{dx^2} < 0\), this indicates that the function has a maximum at \( x = 12 \).