Verified Solution • 2025–26 Session moderate

QJEEM22Maf76c: JEE Main 2022 Mathematics - Class 12 JEEM22Maf76c Solution: Let , . If [a, b] is the range of the function f, then 4a b is…

Let f(x)=2cos⁡−1x+4cot⁡−1x−3x2−2x+10f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10 , x∈[−1,1]x \in [ - 1,1] . If [a, b] is the range of the function f, then 4a −- b is equal to :

Multiple Choice Options

  • A 11
  • B 11 −- π\pi
  • C 11 + π\pi
  • D 15 −- π\pi

Step-by-Step Verified Solution

f(x)=2cos⁡−1x+4cot⁡−1x−3x2−2x+10 ∀x∈[−1,1]f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10\,\forall x \in [ - 1,1]

\Rightarrow f'(x) = - {2 \over {\sqrt {1 - {x^2}} }} - {4 \over {1 + {x^2}}} - 6x - 2 < 0\,\forall x \in [ - 1,1]

So f(x) is decreasing function and range of f(x) is [f(1), f( −- 1)], which is [ π\pi + 5, 5 π\pi + 9]

Now 4a−b=4(π+5)−(5π+9)4a - b = 4(\pi + 5) - (5\pi + 9)

=11−π= 11 - \pi

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