Question

# Find any relative extrema of y=arcsinx - x

Rational functions
Find any relative extrema of $$\displaystyle{y}={\arcsin{{x}}}-{x}$$
$$\displaystyle{y}={\arcsin{{x}}}-{x}$$ is the same as $$\displaystyle{y}={{\sin}^{{-{{1}}}}{\left({x}\right)}}-{x}{\quad\text{and}\quad}{y}'={\left(\sqrt{{{1}-{x}^{{2}}}}\right)}^{{-{{1}}}}-{1}$$. When y'=0 we have a stationary point, so $$\displaystyle\sqrt{{{1}-{x}^{{2}}}}={1},{1}-{x}^{{2}}={1}$$ and x=0, and y=0. When x is small $$\displaystyle{{\sin}^{{-{{1}}}}{\left({x}\right)}}={x}{\quad\text{and}\quad}{{\sin}^{{-{{1}}}}{\left({x}\right)}}=-{{\sin}^{{-{{1}}}}{\left(-{x}\right)}}$$, so y<0 when x<0 and y>0 when x>0 and (0,0) is a point of inflection.
However, $$\displaystyle{{\sin}^{{-{{1}}}}{\left({x}\right)}}={{\sin}^{{-{{1}}}}{\left({180}{\left({2}{n}-{1}\right)}-{x}\right)}}{\quad\text{or}\quad}{{\sin}^{{-{{1}}}}{\left({\left(\pi\right)}{\left({2}{n}-{1}\right)}-{x}\right)}}$$, where n is an integer, so the stationary point is cyclical. This affects f(x) but not f'(x). $$\displaystyle{{\sin}^{{-{{1}}}}{\left({0}\right)}}={180}{n}$$ where n is an integer, so $$\displaystyle{f{{\left({0}\right)}}}={180}{n}{\quad\text{or}\quad}{n}{\left(\pi\right)}$$. The stationary points are therefore "stacked" on the y axis with a separation of 180 degrees or 3.1416 radians, while x ranges from -1 to 1.