# Find any relative extrema of y=arcsinx - x

Find any relative extrema of $y=\mathrm{arcsin}x-x$
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$y=\mathrm{arcsin}x-x$ is the same as $y={\mathrm{sin}}^{-1}\left(x\right)-x\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{y}^{\prime }={\left(\sqrt{1-{x}^{2}}\right)}^{-1}-1$. When y'=0 we have a stationary point, so $\sqrt{1-{x}^{2}}=1,1-{x}^{2}=1$ and x=0, and y=0. When x is small ${\mathrm{sin}}^{-1}\left(x\right)=x\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathrm{sin}}^{-1}\left(x\right)=-{\mathrm{sin}}^{-1}\left(-x\right)$, so y<0 when x<0 and y>0 when x>0 and (0,0) is a point of inflection.
However, ${\mathrm{sin}}^{-1}\left(x\right)={\mathrm{sin}}^{-1}\left(180\left(2n-1\right)-x\right)\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}{\mathrm{sin}}^{-1}\left(\left(\pi \right)\left(2n-1\right)-x\right)$, where n is an integer, so the stationary point is cyclical. This affects f(x) but not f'(x). ${\mathrm{sin}}^{-1}\left(0\right)=180n$ where n is an integer, so $f\left(0\right)=180n\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}n\left(\pi \right)$. 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.