Several Variables
In this section, functions are assumed to be smooth. For a smooth function of several real variables, the condition of being a critical point is equivalent to all of its partial derivatives being zero; for a function on a manifold, it is equivalent to its differential being zero.
If the Hessian matrix at a critical point is nonsingular then the critical point is called nondegenerate, and the signs of the eigenvalues of the Hessian determine the local behavior of the function. In the case of a real function of a real variable, the Hessian is simply the second derivative, and nonsingularity is equivalent to being nonzero. A nondegenerate critical point of a single-variable real function is a maximum if the second derivative is negative, and a minimum if it is positive. For a function of n variables, the number of negative eigenvalues of a critical point is called its index, and a maximum occurs when all eigenvalues are negative (index n, the Hessian is negative definite) and a minimum occurs when all eigenvalues are positive (index zero, the Hessian is positive definite); in all other cases, the critical point can be a maximum, a minimum or a saddle point (index strictly between 0 and n, the Hessian is indefinite). Morse theory applies these ideas to determination of topology of manifolds, both of finite and of infinite dimension.
Read more about this topic: Critical Point (mathematics)
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