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MHT CET202613 April 2026Morning ShiftMathematicsContinuity and DifferentiabilityActual

If f(x) = cases (8 - 2x)^ 1 3 - 2 3 - (243 + 5x)^ 1 5 , & if x 0 k, & if x = 0 cases is continuous at x = 0 , then k =

Options

  1. A5 2
  2. B- 5 2
  3. C27 2
  4. D-27 2

Correct answer

C. 27 2

Step-by-step solution

Since the function f(x) is continuous at x = 0 , we have k = _ x 0 f(x) . k = _ x 0 (8 - 2x)^ 1/3 - 2 3 - (243 + 5x)^ 1/5 This is a 0 0 form. Applying L'Hospital's Rule, we differentiate the numerator and the denominator with respect to x : k = _ x 0 1 3 (8 - 2x)^ -2/3 (-2) - 1 5 (243 + 5x)^ -4/5 (5) Substituting x = 0 into the expression: k = 1 3 (8)^ -2/3 (-2) -(243)^ -4/5 k = - 2 3 1 4 - 1 81 k = - 1 6 - 1 81 k = 81 6 = 27 2

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