`f(x) = arcsinx - 2x` Find any relative extrema of the function

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Borys Shumyatskiy eNotes educator| Certified Educator

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.