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You have shown that KKT is necessary for a local minimum. Also that it is necessary for a local maximum. But you have not shown that a local minimum or local maximum exists. Indeed, there is no local maximum.
So is the KKT point you found a local minimum? That is what 2nd order conditions can assess. The 2nd order (KKT) sufficiency conditions (whose ...
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The notation $C^1$ means $f'$ is continuous (on $\Bbb R$ as the interval is not stated). In general $C^k(a,b]$ means that all of $f',f'',\cdots,f^{(k)}$ are continuous on $(a,b]$.
You are correct that radial unboundedness means that $f\to\infty$ as $\|x\|\to\infty$. This method is essentially that for Lyapunov stability.
Ahmadi and Jungers (2018)1 proved ...
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If a deterministic global optimisation solver (such as Baron) reports a local solution, that solution is reliable. If the solver is terminated prematurely, the global solver will return the best solution it has found so far.
For NLP, it is quite common that global solvers find the global solution very early on, and then spend the majority of time proving it ...
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