The hessian of a function - how to find the maximum or minimum I am taking Calculus 3 and in the last class the professor say that we can use the hessian of a function to see which is the minimum and or maximum of a function...
The problem is that I don't understand exactly what the hessian means, and how I can use it to calculate a minimum or maximum.
Can someone explain me a little bit with a step-by-step example?
Also, How I know that it is definitive positive or definitive negative??
Thanks.
 A: Assume that $f:\ \Omega\to{\mathbb R}^2$ has continuous second partial derivatives in a region $\Omega\subset{\mathbb R}^n$, and that ${\bf p}\in\Omega$ is a critical point of $f$, i.e., that $\nabla f({\bf p})={\bf 0}$. Then Taylor's theorem applied at ${\bf p}$ says that
$$f({\bf p}+{\bf X})-f({\bf p})=\sum_{i, k=1}^n h_{ik}\ X_i\>X_k+\ o\bigl(|{\bf X}|^2\bigr)\qquad({\bf X}\to{\bf 0})\ ,\tag{1}$$
where the symmetric matrix $H:=[h_{ik}]$ given by
$$h_{ik}:={\partial^2 f\over\partial x_i\partial x_k}\biggr|_{\bf p}$$
is the so-called Hessian of $f$ at the critical point ${\bf p}$.
When this Hessian, resp., the quadratic form $$h({\bf X}):=\sum_{i, k=1}^n h_{ik}\ X_i\>X_k\ ,$$ is, say, positive definite, i.e., assumes a positive value at all points ${\bf X}\ne{\bf 0}$, then it assumes a positive minimum $\mu>0$ on the unit sphere $S^{n-1}$, and from $(1)$ it follows that
$$f({\bf p}+{\bf X})-f({\bf p})\geq |{\bf X}|^2\bigl(\mu+o(1)\bigr)\qquad ({\bf X}\to{\bf 0})\ .$$
But this says that for all sufficiently short ${\bf X}\ne{\bf 0}$ we have $f({\bf p}+{\bf X})-f({\bf p})>0$, in other words: that we have a local minimum at ${\bf p}$.
Testing the matrix $H$ for definiteness is a problem of linear algebra, and I won't go into it. When $H$ is only "semidefinite" it is usually difficult to decide whether we have a local extremum at ${\bf p}$. When $H$ is indefinite, i.e., assumes positive and negative values, then we certainly don't have an extremum at ${\bf p}$.
A: Hint: The Hessian is the matrix consisting of second order partial derivatives (compare it to the Jacoby matrix, which is similar, but then for first order partial derivatives). How can you use the second derivative in finding a maximum/minimum?
Additional Hint: The sign of the second derivative tells you whether something is a maximum or a minimum. A negative second deravitive implies a maximum and vice versa. To determine the maximum or minimum, you will also need to make use of the first derivative.
