Question

When finding the critical points of a function, is can there be a point at a value where it makes f(x) undefined? I have x=0 as a critical value, but plugging it into the original function returns and undefined value.

Answer 1

if the function is not defined at some point, it is impossible for the derivative to be defined there

Answer 2

it is of course possible for the derivative to be undefined at a point where the function is defined, for example if \(f(x)=\sqrt{x}\) the \(f(0)=0\) but \(f'(x)=\frac{1}{2\sqrt{x}}\) so
\(f'(0)\) does not exist