Verified Solution • 2025–26 Session moderate

QJEEM21Maf741: JEE Main 2021 Mathematics - Class 12 JEEM21Maf741 Solution: Let M and m respectively be the maximum and minimum values of…

Let M and m respectively be the maximum and minimum values of the function
f(x) = tan −- 1 (sin x + cos x) in [0,π2]\left[ {0,{\pi \over 2}} \right] , then the value of tan(M −- m) is equal to :

Multiple Choice Options

  • A 2+32 + \sqrt 3
  • B 2−32 - \sqrt 3
  • C 3+223 + 2\sqrt 2
  • D 3−223 - 2\sqrt 2

Step-by-Step Verified Solution

Let g(x) = sin x + cos x = 2\sqrt 2 sin (x+π4)\left( {x + {\pi \over 4}} \right)

g(x) ∈\in [1,2]\left[ {1,\sqrt 2 } \right] for x ∈\in [0, π\pi /2]

f(x) = tan −- 1 (sin x + cos x) ∈\in [π4,tan⁡−12]\left[ {{\pi \over 4},{{\tan }^{ - 1}}\sqrt 2 } \right]

tan (tan⁡−12−π4)=2−11+2×2−12−1=3−22({\tan ^{ - 1}}\sqrt 2 - {\pi \over 4}) = {{\sqrt 2 - 1} \over {1 + \sqrt 2 }} \times {{\sqrt 2 - 1} \over {\sqrt 2 - 1}} = 3 - 2\sqrt 2

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