In my lecture the prof wrote that when the derivative does not exist at a point it is also a critical point
I can understand that $f'(c) = 0$ indicates that we have a flat place on our curve, so $f(x)$ is constant there and it is a max or a min, but why is when the derivative not exist also a critical point?
Think of a critical point as a "good candidate" for a point at which a local extremum could occur. To come up with a sensible way of formalizing this, think about common places where local extrema occur: for, say, $f(x)=x^2$, it's where $f'(x)=0$, but for $g(x)=|x|$, it's where $g'(x)$ is undefined (i.e. at $x=0$). We also want a critical point of a function $f(x)$ to be in the domain of $f(x)$, e.g. to avoid calling $0$ a critical point of a function like $f(x)=\frac{1}{x}$: $f'(x)$ is certainly undefined at $x=0$, but $f(0)$ could never be a local extremum as it's not even defined to begin with.