The above link is a portion of a solution to problem 9.31 from Rudin's Principals of Mathematical Analysis. I'm confused as to how the person who solved this jumped from his or her selection of $\delta$ directly led to the fact that the function has a local minimum at $a$. Any clarification would be much appreciated.
2026-04-01 11:57:59.1775044679
Positive definite Hessian matrix yielding a local minimum: detail clarification
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$\delta$ is the radius of the neighborhood of $a$ in $\mathbb{R}^n$ such that $f(x)\ge f(a)$ when $x$ is in that neighborhood. Once you have the inequality in some neighborhood, you have a local minimum.