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JEE Main20207 Jan 2020Evening ShiftMathematicsApplication of DerivativesActual

Let f x be a polynomial of degree 5 such that x = ± 1 are its critical points. If l i m x → 0 2 + f x x 3 = 4 , then which one of the following is not true?

Options

  1. Af is an odd function
  2. Bf 1 - 4 f - 1 = 4 . x = 1 is a point of maximum and x = - 1
  3. Cx = 1 is a point of local minimum and x = - 1 is a point of local maximum
  4. Dx = 1 is a point of local maxima of f

Correct answer

D. x = 1 is a point of local maxima of f

Step-by-step solution

f x = a x 5 + b x 4 + c x 3 Consider, lim x → 0 ⁡ 2 + a x 5 + b x 4 + c x 3 x 3 = 4 ⇒ 2 + c = 4 ⇒ c = 2 Differentiating f x with respect to x f ' x = 5 a x 4 + 4 b x 3 + 6 x 2 = x 2 5 a x 2 + 4 b x + 6 Critical points of the function are given as ± 1 . f ' 1 = 0 ⇒ 5 a + 4 b + 6 = 0 f ' - 1 = 0 ⇒ 5 a - 4 b + 6 = 0 Solving both we get, b = 0 , a = - 6 5 So, f x = - 6 5 x 5 + 2 x 3 And f ' x = - 6 x 4 + 6 x 2 = - 6 x 2 x + 1 x - 1 Hence, local minima at x = - 1

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