Question

Find any relative extrema of y=arcsinx - x

Rational functions
ANSWERED
asked 2020-10-27
Find any relative extrema of \(\displaystyle{y}={\arcsin{{x}}}-{x}\)

Answers (1)

2020-10-28
\(\displaystyle{y}={\arcsin{{x}}}-{x}\) 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.
0
 
Best answer

expert advice

Need a better answer?
...