Question

Why is it considered a saddle point in part (b) if the test fails? http://c86f903f96a3ec2dd206-646e12b456b1588b6ef5c188c33a829b.r15.cf2.rackcdn.com/se13h01033.png

Answer 1

It's probably clear, but just in case these are directions
(a) find the critical points
(b) test for relative extrema
(c) list the critical points for which the Second Partials Test fails

Answer 2

Having that discriminant equal 0 just means that thr test is inconclusive, it doesn't tell you what you can or cannot rule out at that point

Answer 3

I know this test is inconclusive, but in the answer it says it has a saddle point at (0,0,0), where did it get that from?

Answer 4

Have you done multivariable limits yet?

Answer 5

yes

Answer 6

I mean, it's obvious that it's a saddle from the graph, but what confuses me is that it concluded that it was a saddle point in part (b), so it was determined analytically somehow, and I don't see how (probably something stupid I'm not seeing).

Answer 7

Well, when you wanted to determine if a limit didn't exist, you tested paths. If you could show that two paths that converge to the same point had different limits, you could conclude the limit doesn't exist. The same idea can work with saddle points. Your goal this time, though, is to choose two paths that will guarantee different signs. So I want one path to be negative and one path to be positive. This would be a more analytical way to determine if a point is a saddle point. They would've had to have done something along the lines of that.

Answer 8

I see what you're saying, but I highly doubt that we'd have to do something like that in the middle of this problem. Would that even be in the scope of a Calc 3 class? I think I'll just assume it was determined graphically because the more I think about this, the more it seems like it's just a technical thing that's not important to the problem or what I'm supposed to be getting out of it.

Answer 9

Do you agree or do you think I should be able to do it analytically?

Answer 10

Well, it may be something that's mentioned just off on the side in the course. But without having that discriminant test, you're looking at the definition. If you were given the graph prior to seeing the solutions, I suppose you could "eyeball" it. In the end, don't go beyond what you've been taught. If the limit idea seems a little advanced, then don't worry about it :)