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JEE Main202325 Jan 2023Morning ShiftMathematicsApplication of DerivativesActual

Let x = 2 be a local minima of the function f x = 2 x 4 - 18 x 2 + 8 x + 12 , x ∈ - 4 , 4 . If M is local maximum value of the function f in - 4 , 4 , then M =

Options

  1. A12 6 - 33 2
  2. B12 6 - 31 2
  3. C18 6 - 33 2
  4. D18 6 - 31 2

Correct answer

A. 12 6 - 33 2

Step-by-step solution

Given, x = 2 be a local minima of the function f x = 2 x 4 - 18 x 2 + 8 x + 12 , x ∈ - 4 , 4 Now differentiating the above function we get, f ' x = 8 x 3 - 36 x + 8 ⇒ f ' x = 4 2 x 3 - 9 x + 2 Now equating it to zero we get, f ' x = 0 ⇒ 2 x 3 - 9 x + 2 = 0 ⇒ x - 2 2 x 2 + 4 x - 1 = 0 ⇒ x = - 4 ± 24 4 ⇒ x = - 2 ± 6 2 ∴     x = 6 - 2 2 by checking sign change of f ' x for maxima Now rewriting f x = 2 x 4 - 18 x 2 + 8 x + 12 we get, f x = x 2 - 2 x - 9

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