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

Apr 13, 2015

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

For this question the domain of $f \left(x\right) = \sin x - \cos x$ is restricted to $\left(0 , \pi\right)$

$f ' \left(x\right) = \cos x + \sin x$

$\cos x + \sin x = 0$ where $\sin x = - \cos x$ so $\tan x = - 1$ and between $0$ and $\pi$, that occurs at $x = \frac{3 \pi}{4}$

The critical point is $\frac{3 \pi}{4}$.

(An alternative terminology makes critical points ordered pairs. Under this terminology, the critical point would be: $\left(\frac{3 \pi}{4} , \sqrt{2}\right)$