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MHT CET202519 Apr 2025Morning ShiftMathematicsApplication of DerivativesActual

In the mean value theorem, f ^ ( c )= f ( b )- f ( a ) b - a , if a=0, ~b = 1 2 and f (x)=x(x-1)(x-2) , then the value of c is

Options

  1. A1- 15 6
  2. B1- 13 6
  3. C1- 21 6
  4. D1+ 21 6

Correct answer

C. 1- 21 6

Step-by-step solution

The Mean Value Theorem guarantees a c in the interval (0, 1 2 ) such that f'(c) = f( 1 2 ) - f(0) 1 2 - 0 . For f(x) = x(x-1)(x-2) , first compute its derivative: f'(x) = d dx (x^3 - 3x^2 + 2x) = 3x^2 - 6x + 2 . Evaluate the function values: f(0) = 0 f( 1 2 ) = 1 2 (- 1 2 )(- 3 2 ) = 3 8 Then f( 1 2 ) - f(0) 1 2 - 0 = 3 8 - 0 1 2 = 3 4 . Set f'(c) = 3 4 : 3c^2 - 6c + 2 = 3 4 Multiply by 4: 12c^2 - 24c + 8 = 3 12c^2 - 24c + 5 = 0 Solve using the quadratic formula: c = 24 (-24)^2 - 4(12)(5) 2(12) = 24 336 24 = 1 21 6

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