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MHT CET202523 Apr 2025Evening ShiftMathematicsApplication of DerivativesActual

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

Options

  1. A3 4
  2. B6 5
  3. C1
  4. D4

Correct answer

C. 1

Step-by-step solution

Let x and y be two nonzero numbers with x + y = 4 . The sum of reciprocals is S = 1 x + 1 y , which simplifies to S = x + y xy = 4 xy . Since S decreases as xy increases, the minimum of S corresponds to the maximum of xy . By the AM-GM inequality, x + y 2 xy . Substituting x + y = 4 gives 2 xy , so xy 4 with equality when x = y . At x = y = 2 , xy attains its maximum value of 4. The minimum sum of reciprocals is then S = 4 4 = 1 . Minimum value: 1

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