In general, a maximum is a top element, a minimum is a bottom element, and an extremum is either. However, these terms are typically used in analysis, where the order theory is secondary. One also usually speaks of extrema of a function, meaning a top or bottom element of the range of the function under its induced order as a subset of an ordered codomain. In this context, one also considers local extrema of functions; a local extremum of is an extremum of a restriction of to an open subspace of its original domain. In any case, the extremum is strict if the function takes the extreme value only once (in the relevant domain).
We list some sufficient and necessary conditions for a (nice) function on a smooth manifold to have a local extremum at a point . As these are local conditions, we may assume is a function where is an open subset in a Cartesian space . These conditions fall under the rubric of “second derivative test”.
Assume to be a twice-differentiable function, and let in its domain be a critical point: a point where its derivative / Jacobian vanishes. Let be the Hessian matrix of the function. Recall that is a nondegenerate critical point if the symmetric matrix is nondegenerate; equivalently, if is not an eigenvalue of .
Let be a nondegenerate critical point. Then
For to be a strict local minimum within some neighborhood, it is necessary and sufficient that be a positive definite form?.
For to be a strict local maximum within some neighborhood, it is necessary and sufficient that be a negative definite form?.
(The only other possibility left for a nondegenerate critical point is that be an indefinite form?, having a mix of positive and negative eigenvalues. In this case, is a saddle point. For more on this, see Morse theory.)
If is a degenerate critical point (so is an eigenvalue of ), we have:
For to be a local minimum, it is necessary that be a positive semidefinite form?, i.e., that all eigenvalues are nonnegative.
For to be a local maximum, it is necessary that be a negative semidefinite form?, i.e., that all eigenvalues are nonpositive.
These conditions are not sufficient. For a simple example, the origin in is a critical point of , where the Hessian is the zero matrix (hence positive semidefinite and negative semidefinite), but clearly the origin is neither a local maximum nor a local minimum.