Question:
The sum of two nonzero numbers is 4. The minimum value of the sum of their reciprocals is
The sum of two nonzero numbers is 4. The minimum value of the sum of their reciprocals is
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For a fixed sum, the sum of reciprocals is minimized when the numbers are equal.
Updated On: Apr 30, 2026
- \( \frac{3}{4} \)
- \( \frac{6}{5} \)
- 1
- 4
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The Correct Option is C
Solution and Explanation
Step 1: AM-HM Inequality
For positive numbers $x, y$: $\frac{x+y}{2} \ge \frac{2}{1/x + 1/y}$.
Step 2: Substitution
$x+y = 4 \implies \frac{4}{2} \ge \frac{2}{1/x + 1/y}$.
$2 \ge \frac{2}{1/x + 1/y}$.
Step 3: Solve for Reciprocals
$1/x + 1/y \ge 2/2 = 1$.
Step 4: Conclusion
The minimum value is 1 (occurs when $x = y = 2$).
Final Answer:(C)
For positive numbers $x, y$: $\frac{x+y}{2} \ge \frac{2}{1/x + 1/y}$.
Step 2: Substitution
$x+y = 4 \implies \frac{4}{2} \ge \frac{2}{1/x + 1/y}$.
$2 \ge \frac{2}{1/x + 1/y}$.
Step 3: Solve for Reciprocals
$1/x + 1/y \ge 2/2 = 1$.
Step 4: Conclusion
The minimum value is 1 (occurs when $x = y = 2$).
Final Answer:(C)
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