Find two positive numbers whose product is 49 and the sum is minimum.

Answer

To solve the problem of finding two positive numbers whose product is 49 and whose sum is minimized, we can follow these steps:

### Step 1: Define the Variables
Let the two positive numbers be \( x \) and \( y \). According to the problem, we have:
\[ x \cdot y = 49 \]

### Step 2: Express the Sum
We want to minimize the sum of these two numbers:
\[ S = x + y \]
From the product equation, we can express \( y \) in terms of \( x \):
\[ y = \frac{49}{x} \]
Substituting this into the sum equation gives:
\[ S = x + \frac{49}{x} \]

### Step 3: Differentiate the Sum
To find the minimum sum, we differentiate \( S \) with respect to \( x \):
\[ \frac{dS}{dx} = 1 - \frac{49}{x^2} \]

### Step 4: Set the Derivative to Zero
To find critical points, set the derivative equal to zero:
\[ 1 - \frac{49}{x^2} = 0 \]
Solving for \( x \):
\[ \frac{49}{x^2} = 1 \implies x^2 = 49 \implies x = 7 \quad (\text{since } x \text{ must be positive}) \]

### Step 5: Find the Corresponding \( y \)
Now, substitute \( x = 7 \) back into the equation for \( y \):
\[ y = \frac{49}{7} = 7 \]

### Step 6: Calculate the Minimum Sum
Now, we can find the minimum sum:
\[ S = x + y = 7 + 7 = 14 \]

### Conclusion
The two positive numbers are \( 7 \) and \( 7 \), and the minimum sum is \( 14 \).

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