If range of the function f ( x )= sin -1 x - cot -1 x + x 2+2 x +6 is [ a , b ] then [ a + b ] equals [Note: [k] denotes greatest integer less than or equal to k.]

Question

If range of the function f ( x )= sin -1 x - cot -1 x + x 2+2 x +6 is [ a , b ] then [ a + b ] equals [Note: [k] denotes greatest integer less than or equal to k.] (A) 5 (B) 6 (C) 10 (D) 11

Tardigrade Admin · Accepted Answer

f(x)= sin -1 x-((π/2)- tan -1 x)+(x+1)2+5= sin -1 x+ tan -1 x-(π/2)+(x+1)2+5 Domain of f(x) is [-1,1] and f(x) is increasing in [-1,1] f (-1)=(-5 π/4)+5, f (1)=(π/4)+9 ∴ text Range of f ( x ) text is [(-5 π/4)+5, (π/4)+9] ⇒ a + b =-π+14 ⇒[ a + b ]=10

Q. If range of the function f(x)=sin−1x−cot−1x+x2+2x+6 is [a,b] then [a+b] equals [Note : [k] denotes greatest integer less than or equal to k.]

5

6

10

11

f(x)=sin−1x−(2π​−tan−1x)+(x+1)2+5=sin−1x+tan−1x−2π​+(x+1)2+5 Domain of f(x) is [−1,1] and f(x) is increasing in [−1,1] f(−1)=4−5π​+5,f(1)=4π​+9 ∴ Range of f(x) is [4−5π​+5,4π​+9] ⇒a+b=−π+14⇒[a+b]=10

Q. If range of the function $f (x) = sin^{- 1} x - cot^{- 1} x + x^{2} + 2 x + 6$ is $[a, b]$ then $[a + b]$ equals
[Note : [k] denotes greatest integer less than or equal to $k$ .]