the sum of two numbers is fixed. Then its multiplication is maximum, when

Options

A. each number is half of the sum
B. each number is `(1)/(3)` and `(2)/(3)` respectively of the sum
C. each number is `(1)/(4)` and `(3)/(4)` respectively of the sum
D. none of these

Explanation

To solve the problem of finding the maximum multiplication of two numbers whose sum is fixed, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables:**
Let the two numbers be \( A \) and \( B \). According to the problem, their sum is fixed. We can denote this fixed sum as \( S \):
\[ A + B = S \]

2. **Express One Variable in Terms of the Other:**
From the equation above, we can express \( B \) in terms of \( A \):
\[ B = S - A \]

3. **Write the Multiplication Function:**
We want to maximize the product \( P \) of the two numbers:
\[ P = A \times B = A \times (S - A) = AS - A^2 \]

4. **Differentiate the Multiplication Function:**
To find the maximum value, we need to differentiate \( P \) with respect to \( A \):
\[ \frac{dP}{dA} = S - 2A \]

5. **Set the Derivative to Zero:**
To find the critical points, set the derivative equal to zero:
\[ S - 2A = 0 \]
Solving for \( A \):
\[ 2A = S \implies A = \frac{S}{2} \]

6. **Find the Corresponding Value of \( B \):**
Now, substitute \( A \) back into the equation for \( B \):
\[ B = S - A = S - \frac{S}{2} = \frac{S}{2} \]

7. **Conclusion:**
Thus, both numbers \( A \) and \( B \) are equal to \( \frac{S}{2} \). Therefore, the multiplication \( A \times B \) is maximized when both numbers are equal to half of the fixed sum \( S \).

### Final Answer:
The multiplication of the two numbers is maximum when each number is half of the sum.