Find two positive numbers whose sum is 16 and product is maximum.

Answer

To find two positive numbers whose sum is 16 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 16, the second number can be expressed as \( 16 - x \).

### Step 2: Write the Product Function
The product \( P \) of the two numbers can be defined as:
\[ P = x(16 - x) \]
This simplifies to:
\[ P = 16x - x^2 \]

### Step 3: Find the Derivative
To maximize the product, we need to find the derivative of \( P \) with respect to \( x \):
\[ \frac{dP}{dx} = 16 - 2x \]

### Step 4: Set the Derivative to Zero
To find the critical points, set the derivative equal to zero:
\[ 16 - 2x = 0 \]
Solving for \( x \):
\[ 2x = 16 \implies x = 8 \]

### Step 5: Determine the Nature of the Critical Point
To confirm that this value of \( x \) gives a maximum, we can use the second derivative test. The second derivative of \( P \) is:
\[ \frac{d^2P}{dx^2} = -2 \]
Since the second derivative is negative, this indicates that the function has a maximum at \( x = 8 \).

### Step 6: Find the Two Numbers
Now that we have \( x = 8 \), we can find the second number:
\[ 16 - x = 16 - 8 = 8 \]
Thus, the two positive numbers are \( 8 \) and \( 8 \).

### Final Answer
The two positive numbers whose sum is 16 and product is maximum are \( 8 \) and \( 8 \).

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