# What is the difference between a critical point and a stationary point?

$\left({x}_{0} , f \left({x}_{0}\right)\right)$ is a stationary point of $f \left(x\right)$ if $f \left({x}_{0}\right)$ and $f ' \left(x\right)$ exist and is equal to $f ' \left({x}_{0}\right) = 0$
$\left({x}_{0} , f \left({x}_{0}\right)\right)$ is a critical point of $f \left(x\right)$ if $f \left({x}_{0}\right)$ exists and either
$f ' \left({x}_{0}\right)$ does not exist (that is $f \left(x\right)$ is not differentiable at ${x}_{0}$
$f ' \left({x}_{0}\right) = 0$
For example $f \left(x\right) = \sqrt{1 - \frac{1}{{x}^{2} + 1}}$ is not differentiable at $\left(0 , 0\right)$, so $\left(0 , 0\right)$ is a critical point of $f \left(x\right)$ but not a stationary point.