Find two positive numbers whose sum is 14 and product is maximum.

Answer

To solve the problem of finding two positive numbers whose sum is 14 and whose product is maximized, we can follow these steps:

### Step 1: Define the Variables
Let one of the numbers be \( x \). Since the sum of the two numbers is 14, the other number can be expressed as:
\[ y = 14 - x \]

### Step 2: Express the Product
The product \( P \) of the two numbers can be expressed as:
\[ P = x \cdot y = x(14 - x) = 14x - x^2 \]

### Step 3: Differentiate the Product
To find the maximum product, we need to differentiate \( P \) with respect to \( x \):
\[ \frac{dP}{dx} = 14 - 2x \]

### Step 4: Set the Derivative to Zero
To find the critical points, set the derivative equal to zero:
\[ 14 - 2x = 0 \]
Solving for \( x \):
\[ 2x = 14 \implies x = 7 \]

### Step 5: Find the Other Number
Now, substitute \( x = 7 \) back into the equation for \( y \):
\[ y = 14 - x = 14 - 7 = 7 \]

### Step 6: Verify Maximum using Second Derivative Test
To confirm that this critical point gives a maximum, we can use the second derivative test. Differentiate \( \frac{dP}{dx} \):
\[ \frac{d^2P}{dx^2} = -2 \]
Since \( \frac{d^2P}{dx^2} < 0 \), this indicates that the function \( P \) has a maximum at \( x = 7 \).

### Conclusion
The two positive numbers that maximize the product while summing to 14 are:
\[ \text{First number} = 7, \quad \text{Second number} = 7 \]