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My Maths notes don't explain this, but is there a saddle point or a local maximum?

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I assume you're trying to classify a stationary point by looking at the matrix of second derivatives? In that case, you can't draw any conclusion if the matrix is semidefinite, because what happens can be influenced by higher-order terms in the Taylor expansion of the function at the critical point. Investigation of what really happens must be done on a case-by-case basis, using whatever methods one might come up with, and since this goes beyond the simple standard methods, it's usually swept under the rug in introductory calculus courses.

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