critical point

When you hunt for the highest and lowest spots on a landscape, you look where the ground goes flat — but you also keep an eye on cliffs and spikes, where the notion of slope breaks down entirely. Critical points are precisely these suspect locations: the candidates worth examining for extrema.

Definition: a point c in the domain of f is a critical point if either f'(c) = 0 or f'(c) does not exist. The first kind are stationary points (flat tangent); the second kind include corners, cusps, and vertical tangents where no finite derivative is available. The value f(c) at such a point is sometimes called a critical value.

Critical points matter because of a sieve principle: by Fermat's theorem, every interior local extremum of f occurs at a critical point. So when searching for maxima and minima on an interval, it suffices to check the critical points together with the endpoints. But the sieve only narrows the candidates — a critical point need not be an extremum at all (e.g. x = 0 for f(x) = x^3 is critical but is neither a max nor a min).