When a Hessian matrix is negative definite at a critical point then that critical point is a local maximum (Sufficient Condition).
As per the calculus wiki:
Link, when the Hessian is negative semi-definite then, we can only conclude that it is not a local minimum. This seems to suggest that negative semi-definiteness is a necessary condition, not a sufficient one.
Can anyone provide an example of a multiple variable function where we have a negative semi-definite Hessian but not a local maximum? As per my thinking, if we evaluate the hessian to be negative semi-definite at the critical point it must also be a local maximum, but clearly calculus wiki disagrees.
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The simplest example is $-x^3$ in the single variable case, or $-x_1^3-x_2^3$ in the case of two variables. The Hessian matrix is negative semi-definite at $(0,0)$, but there is no maximum at this point.
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$\begingroup$ I see. So it is indeed a necessary condition not a sufficient one. Could you refer me to a method of further inspection to verify that the critical point is indeed a maximum if we have negative semi-definiteness (a non-graphical one). $\endgroup$– KinnoCommented Oct 18, 2021 at 11:48
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$\begingroup$ @Kinno: Yes, the mere finding of a negative semi-definite Hessian does not imply that there is no maximum at this point. Consider $-x_1^4-x_2^4$ for instance, whose Hessian is nsd at $(0,0)$. Hmm, I am not sure that there is a general method allowing you to conclude in all cases. I would recommend to go back to the definition of a maximum and try to study whether $f(x_1,x_2) \leq f(x_1^*,x_2^*)$ for any $(x_1,x_2)$. In our example $-x_1^4-x_2^4 \leq 0$ and so $(0,0)$ corresponds to a global maximum of $f$, even though the Hessian is not negative definite at this point. $\endgroup$– BertrandCommented Oct 18, 2021 at 12:02
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$\begingroup$ Great thank you. I have a feeling this might have to do something with establishing concavity at the intervals on the side of our critical point. Regardless, I will look further into it. $\endgroup$– KinnoCommented Oct 18, 2021 at 12:10