Hello, Chaobunny!
. . . . How?
We have: .
From [2], we have: .
Substitute into [1]: .
We have one critical point: .
We have: .
At , we have: .
Since is negative, there is a saddle point at
So I have and I need to find the maxima and minima. I calculated that , and by setting both to zero I figured out that there are critical points where . Here's where I'm running into problems. I know how to find the x values, but I'm confused about how to find the corresponding y values for these critical points. For the first one, clearly y=0 when x = 0, but I'm not sure of the others because plugging them in to the 2 equations doesn't give me anything definite. In cases like this, what is the best way to find the corresponding y values of the critical points?
I have no idea what you mean by "an obvious corresponding y value". The point is that both equations, and must be true at a critical point. The only solution to is x= 2 (NOT x= -2. If x= -2, , not 0) and, putting that back into the first equation, so that .
The fact that x= 0 satisfies the first equation is irrelevant- it does not satisfy the second equation for any value of y.