#2tan^-1(x)+sin^-1((2x)/(1+x^2))# is independent of #x#, then?
a) #x in [1, oo)#
b) #x in (-oo, -1]#
c) #x in [-1,1]#
d) None of these
a)
b)
c)
d) None of these
1 Answer
We start by noticing that the function
Recall that
So we have two inequalities we must solve, which are
#(2x)/(1 + x^2) ≥ -1#
AND
#(2x)/(1 + x^2) ≤ 1#
Let's solve!
#2x ≥ -x^2 - 1#
#x^2 + 2x + 1 ≥ 0#
This is true on all real numbers as
As for the second, we have:
#2x ≤ x^2 + 1#
#0 ≤ x^2 - 2x + 1#
#0 ≤ (x -1)^2#
This also has a solution of all real numbers since it's a parabola which also opens upwards and has it's minimum on the x-axis.
Therefore the answer is
Hopefully this helps!