Amongest all pairs of positive numbers with product (i) 256 (ii) 64, find those whose sum is least.

Answer

To solve the problem of finding pairs of positive numbers with a given product that minimize their sum, we will follow a systematic approach for both cases (i) product = 256 and (ii) product = 64.

### Case (i): Product = 256

1. **Define the Variables**:
Let the two positive numbers be \( x \) and \( y \).
Given that \( xy = 256 \), we can express \( y \) in terms of \( x \):
\[ y = \frac{256}{x} \]

2. **Express the Sum**:
The sum \( S \) of the two numbers is:
\[ S = x + y = x + \frac{256}{x} \]

3. **Differentiate the Sum**:
To find the minimum sum, we differentiate \( S \) with respect to \( x \):
\[ S' = 1 - \frac{256}{x^2} \]

4. **Set the Derivative to Zero**:
To find critical points, set the derivative equal to zero:
\[ 1 - \frac{256}{x^2} = 0 \]
\[ \frac{256}{x^2} = 1 \implies x^2 = 256 \implies x = 16 \quad (\text{since } x \text{ is positive}) \]

5. **Find the Corresponding \( y \)**:
Substitute \( x = 16 \) back to find \( y \):
\[ y = \frac{256}{16} = 16 \]

6. **Calculate the Minimum Sum**:
The minimum sum is:
\[ S = 16 + 16 = 32 \]

### Case (ii): Product = 64

1. **Define the Variables**:
Let the two positive numbers be \( x \) and \( y \).
Given that \( xy = 64 \), we can express \( y \) in terms of \( x \):
\[ y = \frac{64}{x} \]

2. **Express the Sum**:
The sum \( S \) of the two numbers is:
\[ S = x + y = x + \frac{64}{x} \]

3. **Differentiate the Sum**:
To find the minimum sum, we differentiate \( S \) with respect to \( x \):
\[ S' = 1 - \frac{64}{x^2} \]

4. **Set the Derivative to Zero**:
Set the derivative equal to zero:
\[ 1 - \frac{64}{x^2} = 0 \]
\[ \frac{64}{x^2} = 1 \implies x^2 = 64 \implies x = 8 \quad (\text{since } x \text{ is positive}) \]

5. **Find the Corresponding \( y \)**:
Substitute \( x = 8 \) back to find \( y \):
\[ y = \frac{64}{8} = 8 \]

6. **Calculate the Minimum Sum**:
The minimum sum is:
\[ S = 8 + 8 = 16 \]

### Summary of Results:
- For the product of 256, the pair is \( (16, 16) \) with a minimum sum of 32.
- For the product of 64, the pair is \( (8, 8) \) with a minimum sum of 16.