`f(x) = arcsinx - 2x` Find any relative extrema of the function
This function is defined on `[-1, 1]` and is differentiable on `(-1, 1).` Its derivative is `f'(x) = 1/sqrt(1-x^2) - 2.`
The derivative doesn't exist at `x = +-1.` It is zero where `1-x^2 = 1/4,` so at `x = +-sqrt(3)/2.` It is an even function and it is obviously increases for positive x and decreases for negative x. Hence it is positive on `(-1, -sqrt(3)/2) uu (sqrt(3)/2, 1)` and negative on `(-sqrt(3)/2, sqrt(3)/2),` and the function `f` increases and decreases respectively.
This way we can determine the maximum and minimum of `f:` `-1` is a local (one-sided) minimum, `1` is a local one-sided maximum, `-sqrt(3)/2` is the local maximum and `sqrt(3)/2` is a local minimum.