Question

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

Answer 1

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.

Answer 2

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.

Answer 3

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