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MHT CET202619 April 2026Morning ShiftMathematicsApplication of DerivativesActual

The value of c satisfied by the Rolle's theorem for the function f(x) = x^2(1 - x)^2 , x [0, 1] is...

Options

  1. A0
  2. B1
  3. C1 2
  4. D-1

Correct answer

C. 1 2

Step-by-step solution

Given function is f(x) = x^2(1 - x)^2 on the interval [0, 1] . The function is a polynomial, so it is continuous on [0, 1] and differentiable on (0, 1) . Also, f(0) = 0 and f(1) = 0 , so f(0) = f(1) . By Rolle's theorem, there exists at least one c (0, 1) such that f'(c) = 0 . Differentiating f(x) with respect to x : f'(x) = d dx [x^2(1 - x)^2] = 2x(1 - x)^2 + x^2 2(1 - x)(-1) f'(x) = 2x(1 - x)[(1 - x) - x] = 2x(1 - x)(1 - 2x) Setting f'(c) = 0 : 2c(1 - c)(1 - 2c) = 0 The possible values are c = 0 , c = 1 , and c =

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