show that f (x) = 2x + cot inverse x + log (root of 1+ x square -x) is increasing - Maths - Continuity and Differentiability

Question

show that f (x) = 2x + cot inverse x + log (root of 1+ x square -x)
is increasing in R

Rishabh Mittal · Accepted Answer

fx=2x+cot-1x+log1+x2-xOn differentiating wrt x , we getf'x=2-11+x2+11+x2-xd1+x2-xdx      =2-11+x2+11+x2-x2x21+x2-1      =2-11+x2+11+x2-xx-1+x21+x2      =2-11+x2-11+x2=2-1+1+x21+x2>0   since 1+1+x21+x2<2So , fx is increasing in R

show that f (x) = 2x + cot inverse x + log (root of 1+ x square -x) is increasing in R.