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MHT CET202620 April 2026Evening ShiftMathematicsApplication of DerivativesActual

The function f(x) = x(x + 3)e^ - ( 1 2 )x satisfies all the conditions of Rolle's theorem in [-3, 0] , then c =

Options

  1. A-3
  2. B-2
  3. C-1
  4. D0

Correct answer

B. -2

Step-by-step solution

Given f(x) = (x^2 + 3x)e^ - x 2 Differentiating with respect to x : f'(x) = (2x + 3)e^ - x 2 + (x^2 + 3x)e^ - x 2 (- 1 2 ) f'(x) = e^ - x 2 ( 2x + 3 - x^2 + 3x 2 ) f'(x) = e^ - x 2 ( -x^2 + x + 6 2 ) By Rolle's theorem, there exists c (-3, 0) such that f'(c) = 0 . -c^2 + c + 6 2 = 0 c^2 - c - 6 = 0 (c - 3)(c + 2) = 0 c = 3 or c = -2 Since c (-3, 0) , we get c = -2 . Answer: -2

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