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ncf
2017-02-06T11:25:15.726Z
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www.uio.no

By late tonight, please send me suggestions for whatever of the problems to cover tomorrow. I'll look into it. So short in advance, I will certainly have to just make an executive decision.

The end of the problem:

We found a threshold value, which I denote as t, which was 288/23. It had the following properties: if (and only if!) k>t, then positive definite; if k<t, then indefinite. We know that the form is not negative semidefinite, so for k=t there are only two possibilities: indefinite, or positive semidefinite.
The following shows positive semidefiniteness at the threshold level k=t:

k is a parameter in the function, and Q depends continuously on k.

For any k>t, it is so that Q(x,y,z)>0 for every non-null (x,y,z); that is positive definiteness (which we have proven!)

Fix (x,y,z). Then Q(x,y,z)=q(k) depends on k only. For k>t, q(k)>0. Let k->t from above. Then q...