please answer it

Question

if x is greater than equal to 0 and A=arcsinx+arccosx-arctanx then find range of A

Answer 1

A = Arcsinx + Arccosx - Arctanx

 

The domain of Arcsinx is [-1,1]

 

      domain of Arccosx is [-1,1]

 

      domain of Arctanx is (-∞,∞)

 

The domain of A is the intersection of the domains of the three inverse trig functions = [-1,1]

 

But, by assumption, x≥0.  So, we need only consider values of x that lie in the interval [0,1].  

 

If x doesn't equal 1, then dA/dx = 1/√(1-x²) - 1/√(1-x²)  - 1/(1+x²) = -1/(1+x²) < 0.

 

  So, A is continuous and decreasing on the interval [0,1]. Therefore, the maximum value of A occurs at the left endpoint of the interval [0,1] and the minimum occurs at the right endpoint.  Since A is continuous on the interval, A takes on all values between the max and min.   

 

     Maximum value of A = Arcsin0 + Arccos0 - Arctan0 = 0 + π/2 - 0 = π/2

 

     Minimum value of A = Arcsin1 + Arccos1 - Arctan1 = π/2 + 0 - π/4 = π/4

 

Range of A = [π/4, π/2]