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JEE Main202226 Jul 2022Evening ShiftMathematicsApplication of DerivativesActual

If the maximum value of a , for which the function f a x = tan - 1 2 x - 3 a x + 7 is non-decreasing in - π 6 , π 6 , is a ¯ , then f a ¯ π 8 is equal to

Options

  1. A8 - 9 π 4 9 + π 2
  2. B8 - 4 π 9 4 + π 2
  3. C8 1 + π 2 9 + π 2
  4. D8 - π 4

Correct answer

A. 8 - 9 π 4 9 + π 2

Step-by-step solution

Given f a x = tan - 1 2 x - 3 a x + 7 is non-decreasing function in - π 6 , π 6 , so f a ' x = 2 1 + 4 x 2 - 3 a ≥ 0 i.e. a ≤ 2 3 1 + 4 x 2 min Now, maximum value of a is at x = 0 i.e. a max = a ¯ = 6 9 + π 2 Hence, f a ¯ π 8 = tan - 1 π 4 - 3 6 9 + π 2 π 8 + 7 = tan - 1 π 4 - 9 π 4 π 2 + 9 + 7 = 8 - 9 π 4 π 2 + 9

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