The correct option is D increases in (−∞, ∞)
f(x)=2x+cot−1 x+log (√1+x2−x)∴f'(x)=2−11+x2+1√1+x2−x(x√1+x2−1)⇒f'(x)=1+2x21+x2−1√1+x2⇒f'(x)=1+2x21+x2−√1+x21+x2⇒f'(x)=x2+√1+x2(√1+x2−1)1+x2Hence, f'(x)>0 for all x.Hence, f(x) is an increasing function in(−∞,∞) and, in particular f(x) is an increasing function in (0,∞).