Question

If `f(x)=sinx-cosx`, what are the critical points on the interval (0,pi)?

Answer 1

Critical points are elements of the domain at which `f'(x) = 0` or `f'(x)` does not exist.

For this question the domain of `f(x) = sinx-cosx` is restricted to `(0, pi)`

`f'(x) = cos x +sinx`

`cos x +sinx = 0` where `sinx=-cosx` so `tanx = -1` and between `0` and `pi`, that occurs at `x= (3 pi)/4`

The critical point is `(3 pi)/4`.

(An alternative terminology makes critical points ordered pairs. Under this terminology, the critical point would be: `( (3 pi)/4, sqrt2)`