Show that `f(x) = 2x + cot^-1 x + log(sqrt(1+x^2)-x)` is increasing in `R`

Options

A. increases in `[0,oo)`
B. idecreases in `[0,oo)`
C. neither increases nor decreases in `[0,oo)`
D. increases in `(-oo,oo)`

Answer

1,4

Explanation

We have
f(x)=`2x+cot^(-1)x+logsqrt(1+x^(2)-x)`
`therefore f(X)=2(1)/(1+x^(2))+(1)/(sqrt(1+x^(2))-x)(x)/(sqrt(1+x^(2))-1)`
`=(1+2x^(2))/(1+x^(2))-(1)/sqrt(1+x^(2))=(1+32x^(2))/(1+x^(2))=-sqrt(1+x^(2))/(1+x^(2))`
`=(x^(2)+sqrt(1+x^(2))sqrt(1+x^(2))-1)/(1+x^(2))gt0` for all x
Hence f(X) is an increasing function in `(-oo,oo)` and in particular in `(0,oo)`