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Let $f:\mathbb{R}^n\to\mathbb{R}$ be a continuous differentiable function with a unique stationary point at $x_0\in\mathbb{R}^n$. If $x_0$ is a local minimum, is it necessarily the global minimum? (I know the answer is yes for $n=1$.)

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A critical point can be a local minimum, a local maximum, a saddle point or none of these depending on the positive/negative definiteness of the function's Hessian at the critical point (refer to this link for more information: https://en.wikipedia.org/wiki/Second_partial_derivative_test).

However, it is known that each local optimum of a convex function f(x) over a convex set D is also a global optimum over D. A proof of this fundamental property of convex optimization can be found in several articles and books on convex programming, e.g. Bertsekas Optimization book, Mikulas Luptacik's book on optimization and economic analysis, and Boyd and Vandenberghe' "Convex optimization book that can be found in this URL: https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf

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    $\begingroup$ This answer is quite sparse; you should elaborate or expect it to be deleted. $\endgroup$
    – Integrand
    Oct 12, 2020 at 22:52
  • $\begingroup$ Thank you for your comment. I did elaborate as per the above $\endgroup$ Oct 13, 2020 at 18:57

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