study a function defined by f(x)=Arccos[(1-x^2)/(1+x^2)]?
The domain of $\displaystyle f(x)=\arccos\left(\frac{1-x^2}{1+x^2}\right)$ consists of all values of $\displaystyle x$ such that
$\displaystyle -1\le \frac{1-x^2}{1+x^2}\le 1.$
I found a different answer for both the domain and $\displaystyle f'(x)$. Hint: since $\displaystyle 1+x^2$ is positive, multiplying the three expressions by $\displaystyle 1+x^2$ will preserve the inequalities and result in an equivalent statement.
Extrema are found only at critical points, i.e., boundary points, points at which $\displaystyle f'(x)=0$, and points at which $\displaystyle f'(x)$ doesn't exist.
Inflection points are found where $\displaystyle f''(x)=0$, $\displaystyle f''(x)<0$ on one side next to the point, and $\displaystyle f''(x)>0$ on the other side next to the point.