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JEE MainMathematicsApplication of Derivatives

Consider the function f(x) = e^x x^2 - x + 1 . If M is the local maximum value of f(x) and m is the local minimum value of f(x) , then the ratio M m is equal to:

Options

  1. Ae 3
  2. Be^3 3
  3. C3 e
  4. D1 3

Correct answer

C. 3 e

Step-by-step solution

Given f(x) = e^x x^2 - x + 1 . Differentiating with respect to x using the quotient rule: f'(x) = e^x(x^2 - x + 1) - e^x(2x - 1) (x^2 - x + 1)^2 f'(x) = e^x(x^2 - 3x + 2) (x^2 - x + 1)^2 To find the critical points, set f'(x) = 0 . Since e^x > 0 and the denominator is always positive (as its discriminant = 1 - 4 = -3 x^2 - 3x + 2 = 0 (x - 1)(x - 2) = 0 The critical points are x = 1 and x = 2 . Analyzing the sign of f'(x) : For x 0 (function is increasing). For 1 For x > 2 , f'(x) > 0 (function is increasing). By th

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