Critical Point (mathematics) - Optimization

Optimization

By Fermat's theorem, local maxima and minima of a function can occur only at its critical points. However, not every stationary point is a maximum or a minimum of the function — it may also correspond to an inflection point of the graph, as for ƒ(x) = x3 at x = 0, or the graph may oscillate in the neighborhood of the point, as in the case of the function defined by the formulae ƒ(x) = x2sin(1/x) for x ≠ 0 and ƒ(0) = 0, at the point x = 0.