Critical point of two variable function

[gertrudethegr]

Homework Statement

Each of the functions f have a critical point at (0,0), however at this point the second derivatives are all zero. Determine te type of critical point at (0,0) in this case

1) f(x,y)=x²y+xy²
2)f(x,y) = x⁴+2*x³y+x²y²+y⁴

The Attempt at a Solution

For the first part by plotting it on Wofram I saw that it was a saddle, and i substituted the values x=y and x=-y to show that there is a saddle point in one direction and a straight line in the other, hence a saddle point, is this sufficient

On 2) I know I must use taylors expansion, but up to how many terms, and what do I do once i get there?

 

[Mark44]

Homework Statement

Each of the functions f have a critical point at (0,0), however at this point the second derivatives are all zero. Determine te type of critical point at (0,0) in this case

1) f(x,y)=x²y+xy²
2)f(x,y) = x⁴+2*x³y+x²y²+y⁴

The Attempt at a Solution

For the first part by plotting it on Wofram I saw that it was a saddle, and i substituted the values x=y and x=-y to show that there is a saddle point in one direction and a straight line in the other, hence a saddle point, is this sufficient

I doubt it very much. I don't see anything in your work that indicates you took partial derivatives. Your textbook should have some examples of categorizing critical points by the use of partials.

On 2) I know I must use taylors expansion, but up to how many terms, and what do I do once i get there?

I don't think a Taylor series has anything to do with this problem. The comment I made before applies here as well.

 

[gertrudethegr]

The only examples I found using partials considered the second derivative at the given point, however as the questions states the second derivative at the points are all 0, hence all the examples I have found give an inconclusive solution... that is why I have become unstuck, but if I take third derivatives at the point (0,0) there are some non zero solutions, but I am unsure how to identify them as max, min, saddle...