If the sum of two +ve numbers is 18 then the maximum value of their product is

Options

A. 8,10
B. 9,9
C. 6,12
D. None of these

Answer

To find the maximum value of the product of two positive numbers whose sum is 18, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables**:
Let the two positive numbers be \( x \) and \( y \).

2. **Set Up the Equation**:
According to the problem, the sum of the two numbers is given:
\[ x + y = 18 \]

3. **Express One Variable in Terms of the Other**:
From the equation \( x + y = 18 \), we can express \( y \) in terms of \( x \):
\[ y = 18 - x \]

4. **Formulate the Product**:
The product \( P \) of the two numbers can be expressed as:
\[ P = x \cdot y = x(18 - x) = 18x - x^2 \]

5. **Find the Derivative**:
To maximize the product, we need to find the derivative of \( P \) with respect to \( x \):
\[ \frac{dP}{dx} = 18 - 2x \]

6. **Set the Derivative to Zero**:
To find the critical points, set the derivative equal to zero:
\[ 18 - 2x = 0 \]

7. **Solve for \( x \)**:
Rearranging gives:
\[ 2x = 18 \implies x = 9 \]

8. **Find \( y \)**:
Substitute \( x = 9 \) back into the equation for \( y \):
\[ y = 18 - x = 18 - 9 = 9 \]

9. **Calculate the Maximum Product**:
Now, substitute \( x \) and \( y \) back into the product:
\[ P = x \cdot y = 9 \cdot 9 = 81 \]

10. **Verify Maximum with Second Derivative**:
To confirm that this is a maximum, we can check the second derivative:
\[ \frac{d^2P}{dx^2} = -2 \]
Since the second derivative is negative, this confirms that \( P \) has a maximum at \( x = 9 \).

### Final Answer:
The maximum value of the product of the two numbers is \( 81 \).

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