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MHT CET202522 Apr 2025Morning ShiftMathematicsApplication of DerivativesActual

If f (x)=x e ^ x(1-x) , then f (x) is

Options

  1. Aincreasing in R
  2. Bincreasing in (- 1 2 , 1 )
  3. Cdecreasing in R
  4. Ddecreasing in [- 1 2 , 1 ]

Correct answer

B. increasing in (- 1 2 , 1 )

Step-by-step solution

The derivative of f(x) = x e^ x-x^2 is found using the product rule, with u(x) = x and v(x) = e^ x-x^2 . Since u'(x) = 1 and v'(x) = e^ x-x^2 (1-2x) by the chain rule, we obtain the derivative: f'(x) = e^ x-x^2 + x e^ x-x^2 (1-2x) = e^ x-x^2 (1 + x - 2x^2) = -e^ x-x^2 (2x^2 - x - 1) . The exponential term e^ x-x^2 is always positive, so the sign of f'(x) depends on -(2x^2 - x - 1) . Solving 2x^2 - x - 1 = 0 gives x = 1 1+8 4 = 1 3 4 , yielding roots at x = - 1 2 and x = 1 . For f'(x) > 0 , we require 2x^2 - x - 1 T

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