Question

The sum of two nonzero numbers is 4. The minimum value of the sum of their reciprocals is

• A. \( \frac{3}{4} \)
• B. \( \frac{6}{5} \)
• C. 1
• D. 4

Hint:

For a fixed sum, the sum of reciprocals is minimized when the numbers are equal.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the sum of two numbers and need to minimize the sum of their reciprocals.
Step 2: Key Formula or Approach:
Let the numbers be \( x \) and \( y \). We are given \( x + y = 4 \).
We need to minimize \( S = \frac{1}{x} + \frac{1}{y} \).
We can use the AM-HM inequality: \( \frac{x+y}{2} \ge \frac{2}{1/x + 1/y} \).
Step 3: Detailed Explanation:
From AM-HM inequality:
\[ \frac{x+y}{2} \ge \frac{2}{1/x + 1/y} \] \[ \frac{4}{2} \ge \frac{2}{1/x + 1/y} \] \[ 2 \ge \frac{2}{S} \] \[ S \ge 1 \] The equality holds when \( x = y \).
Since \( x + y = 4 \), we have \( x = y = 2 \).
The minimum value of the sum of reciprocals is \( \frac{1}{2} + \frac{1}{2} = 1 \).
Step 4: Final Answer:
The minimum value is 1.