The sum of the two numbers is 8, what will be the minimum value of the sum of their reciprocals.

Answer

To solve the problem of finding the minimum value of the sum of the reciprocals of two numbers whose sum is 8, we can follow these steps:

### Step 1: Define the Variables
Let the two numbers be \( x \) and \( y \). According to the problem, we have:
\[ x + y = 8 \]

### Step 2: Express One Variable in Terms of the Other
From the equation above, we can express \( y \) in terms of \( x \):
\[ y = 8 - x \]

### Step 3: Write the Function for the Sum of Reciprocals
We need to find the minimum value of the sum of their reciprocals, which can be expressed as:
\[ f(x) = \frac{1}{x} + \frac{1}{y} \]
Substituting \( y \) from Step 2:
\[ f(x) = \frac{1}{x} + \frac{1}{8 - x} \]

### Step 4: Differentiate the Function
To find the minimum, we first differentiate \( f(x) \):
\[ f'(x) = -\frac{1}{x^2} + \frac{1}{(8 - x)^2} \]

### Step 5: Set the Derivative to Zero
To find the critical points, we set the derivative equal to zero:
\[ -\frac{1}{x^2} + \frac{1}{(8 - x)^2} = 0 \]
This simplifies to:
\[ \frac{1}{x^2} = \frac{1}{(8 - x)^2} \]

### Step 6: Cross Multiply and Simplify
Cross multiplying gives us:
\[ (8 - x)^2 = x^2 \]
Expanding the left side:
\[ 64 - 16x + x^2 = x^2 \]
This simplifies to:
\[ 64 - 16x = 0 \]

### Step 7: Solve for \( x \)
Solving for \( x \):
\[ 16x = 64 \implies x = 4 \]

### Step 8: Find the Corresponding \( y \)
Using the value of \( x \) to find \( y \):
\[ y = 8 - x = 8 - 4 = 4 \]

### Step 9: Verify the Nature of the Critical Point
To confirm that this critical point is a minimum, we calculate the second derivative:
\[ f''(x) = \frac{2}{x^3} + \frac{2}{(8 - x)^3} \]
Evaluating at \( x = 4 \):
\[ f''(4) = \frac{2}{4^3} + \frac{2}{(8 - 4)^3} = \frac{2}{64} + \frac{2}{64} = \frac{1}{32} + \frac{1}{32} = \frac{1}{16} > 0 \]
Since \( f''(4) > 0 \), this confirms that \( x = 4 \) is indeed a point of minimum.

### Step 10: Calculate the Minimum Value of the Function
Finally, we substitute \( x = 4 \) back into the function to find the minimum value of the sum of the reciprocals:
\[ f(4) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \]

### Conclusion
The minimum value of the sum of the reciprocals of the two numbers is:
\[ \boxed{\frac{1}{2}} \]

---