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. An example of this would be $f(x) = -x^3$
Now if we extend this to multiple variables, and suppose we end up with a negative semi-definite Hessian evaluated at the critical point. In that case, we can only rule out the possibility of the critical point being a local minimum. We still cannot say anything about it being a local max.
Is there a general procedure to conclusively establish if our critical point is a local maximum if the Hessian is negative semi-definite, because it may very well be the case that it is not.