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JEE Main202127 Aug 2021Evening ShiftMathematicsApplication of DerivativesActual

Let M and m respectively be the maximum and minimum values of the function f ( x ) = tan - 1 ( sin x + cos x ) in 0 , π 2 . Then the value of tan ( M - m ) is equal to:

Options

  1. A2 - 3
  2. B3 - 2 2
  3. C3 + 2 2
  4. D2 + 3

Correct answer

B. 3 - 2 2

Step-by-step solution

Given, the function, f ( x ) = tan - 1 ( sin x + cos x ) in 0 , π 2 f ' x = cos x - sin x 1 + sin x + cos x 2 = cos x - sin x 1 + sin 2 x f ' x = 0 cos x - sin x = 0 tan x = 1 x = π 4 Then, 1 ≤ sin x + cos x ≤ 2                   x ∈ π 4 , tan - 1 2 M = tan - 1 ( 2 ) m = tan - 1 ( 1 ) M - m = tan - 1 2 - tan - 1 1 M - m = tan - 1 2 - 1 1 + 2 M - m = tan - 1 ( 3 - 2 2 ) tan ( M - m ) = 3 - 2 2

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