why hessian is applied to identify the characteristcs of stationary points of f(x,y)?

Question

anonymous · Answer

The Hessian captures the curvature of the graph (a surface in $\mathbb{R}^{n+1}$) information in an $n$-variable real function; this is apparent from Taylor's theorem.

anonymous · Answer

In the same way you use the second derivative in single-variable calculus, the Hessian describes how our function contours around a point and thus potentially allowing us to determine whether we have a minimum, maximum, or saddle point.

anonymous · Answer

https://en.wikipedia.org/wiki/Taylor_expansion#Taylor_series_in_several_variables