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MHT CET202526 Apr 2025Evening ShiftMathematicsApplication of DerivativesActual

The minimum value of a x+ by where x y = c ^2 is

Options

  1. A2 c a b
  2. B2 a b c
  3. C-2 c a b
  4. D2 c(a b)

Correct answer

A. 2 c a b

Step-by-step solution

Given E = ax + by with constraint xy = c^2 . Expressing y = c^2 x , we obtain E(x) = ax + bc^2 x . Differentiating yields E'(x) = a - bc^2 x^2 . Setting E'(x) = 0 gives x^2 = bc^2 a , so x = c b a . For x = c b a , y = c a b , and E = 2c ab . For x = -c b a , y = -c a b , and E = -2c ab . The second derivative E''(x) = 2bc^2 x^3 shows the first gives a local minimum and the second a local maximum. Since E(x) - as x 0^- or x - , no global minimum exists, but the minimal local extremum is -2c ab . Final answer: C

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