Question

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

Answer 1

`(x_0,f(x_0))` is a stationary point of `f(x)` if `f(x_0)` and `f'(x)` exist and is equal to `f'(x_0)=0`

`(x_0,f(x_0))` is a critical point of `f(x)` if `f(x_0)` exists and either
`f'(x_0)` does not exist (that is `f(x)` is not differentiable at `x_0`
or
`f'(x_0) = 0`

For example `f(x) = sqrt(1-1/(x^2+1))` is not differentiable at `(0,0)`, so `(0,0)` is a critical point of `f(x)` but not a stationary point.
graph{sqrt(1-1/(x^2+1)) [-2.434, 2.434, -1.215, 1.218]}