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JEE Main202621 January 2026Evening ShiftMathematicsDifferentiationActual

Let f(x)=x³+x² f^ (1)+2 x f^ (2)+f^ (3), x R . Then the value of f^ (5) is:

Options

  1. A117 5
  2. B657 5
  3. C2 5
  4. D62 5

Correct answer

A. 117 5

Step-by-step solution

Let a = f'(1) , b = f''(2) , c = f'''(3) . Then f(x) = x^3 + ax^2 + 2bx + c . Taking derivatives: f'(x) = 3x^2 + 2ax + 2b , f''(x) = 6x + 2a , f'''(x) = 6 . From f'(1) = a : 3 + 2a + 2b = a gives a = -3 - 2b . From f''(2) = b : 12 + 2a = b . From f'''(3) = c : c = 6 . Substituting the first into the second: b = 12 + 2(-3 - 2b) = 6 - 4b gives 5b = 6 , so b = 6 5 . Then a = -3 - 12 5 = - 27 5 . f(x) = x^3 - 27 5 x^2 + 12 5 x + 6 f'(x) = 3x^2 - 54 5 x + 12 5 f'(5) = 75 - 54 + 12 5 = 117 5

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