Question

How would I be able to classify this:
Consider the function f(x,y) = x^{5}y - 35 x^{4} + 16807
y. Find and classify the critical point of the function.

There is a critical point at
(, )

Answer 1

CP at (-7,-4)

Answer 2

So to classify it, you basically are going to be seeing if it's a max, min, or saddle point, since those are your only possibilities. Remember in cal 1 how you did it by looking at concavity which was just the second derivative? Except this time it's a little different.

Answer 3

Thinking its related to the 2nd partials test

Answer 4

Yes?

Answer 5

That's it, yeah.

Answer 6

If it wouldn't bother, fyy=0? I'm unsure

Answer 7

Well it's more in depth than that, it's really:

\[f_{ x x}f_{yy}-f_{xy}^2\] So if you take this formula and plug in your point (x,y) and get a positive value, then it's a max or a min. If you get a negative number then you know it's a saddle point. If 0, you have to do something else to figure out what it is.

If that value is larger than 0 I said it could be max or min, which means to determine that, just look at fxx or fyy since either of those will tell you if it's concave up or down just like you've done in the past.

Answer 8

http://tutorial.math.lamar.edu/Classes/CalcIII/RelativeExtrema.aspx

Answer 9

No no i figured that out, but is fyy=0? I already know what d(x,y) formula is

Answer 10

And i know the chart related to the situation.

Answer 11

If there's no y at all, its 0 isn't it?

Answer 12

Yeah, fyy=0.

Answer 13

Huh

Answer 14

Nothing mysterious going on here, it's a constant with respect to y, so its derivative must be 0, plain and simple.

Answer 15

And d is suppose to be rather large right?

Answer 16

eh, really -ve, but i think is so

Answer 17

D isn't supposed to be anything in general. In this case, d=-5x.

So plug in the point (-7,-4) and you get d=35 which is indeed larger than 0 meaning it's either a max or a min point.