My basis on differential manifolds calculus and differential geometry being very superficial, I'm trying to understand this section on WP's article.
I'm not being able to realize why most of the statements are true. For instance:
then checking whether all the eigenvalues of the Hessian $H_xf$ are negative
Why is this necessary for the point to be a maximum (if that's the case)?
And if it wouldn't be too broad or hard to do on this format, any development of the subject (of the section) in a way that would help me gain intuitivity about this issue would be great!
Sorry if the question is poorly asked, I couldn't find another way to do it.
Thank you!
The Hessian of a function is a generalization of the 2nd derivative of a single-variable function $f(x)$. If $x=a$ is a critical point of $f(x)$, a sufficient criterion for $a$ be a maximum is that the second derivative of $f$ at $a$ be negative: $f''(a) < 0$ (https://en.wikipedia.org/wiki/Derivative_test#Second_derivative_test).
For multivariable functions, "second derivative" generalizes to "Hessian," and "negative (scalar)" generalizes to "negative definite (linear transformation)". A linear transformation $A$ is said to be negative definite if $-A$ is positive-definite. Positive-definiteness can be characterized using eigenvalues: https://en.wikipedia.org/wiki/Positive-definite_matrix#Characterizations.
To get comfortable with manifolds, I would recommend the relevant sections in "Ordinary Differential Equations" by V. Arnol'd and, for a more thorough calculus context, "Mathematical Analysis" by Zorich (both books are published by UniversiText).