Happy new year guys! Don't ask me why I'm doing maths on New Years Eve

Question:

Find the stationary points and examine the following function for relative extrema and saddle points:

which implies x=0 or

substituting x=0 gives you end up getting which is not defined. substituting gives you which implies y=0.

So (1,0) is a stationary point - I am able to find the nature of this point, so don't worry about helping me with that.

From the equation, you get , so , substituting into the equation, you get

So is another stationary point. Again I'm able to find the nature of this point.

These seem right because I've substituted them into the and equations and they work.

My concern is there must be more, because looking at the graph, there seem to be at least 2 maximum points and a minimum. Have I gone wrong somewhere or missed something? Thanks for taking the time to look at this.