Classifying Critical Points So let’s say we’ve got a critical
point of a multivariable function . That is, a point where the
differential vanishes. We want something like the second
derivative test that might tell us more about the behavior of
the function near that point, and to identify (some) local
maxima and minima. We’ll assume here that is twice continuously
differentiable in some region around . The analogue of the
second derivative for multivariable functions is the second
differential . This function assigns to every point a bilinear
function of two displacement vectors and , and it measures the
rate at which the directional derivative in the direction of is
changing as we move in the direction of . That is, If we choose
coordinates on given by an orthonormal basis , we can write the
second differential in terms of coordinates This matrix is often
called the “Hessian” of at the point . As I said above, this is
a bilinear form. Further, Clairaut’s theorem tells us that it’s
a symmetric form. Then the