regular surface- solution check

For which values of , is a regular surface , where

is defined as

I found 2 theorems I think I should use when solving this problem

thm 1

is a regular value of the function iff the partial derivatives do not disappear instantly for any point of the pre-image

thm 2

If is differentiable and is a regular value , then is a regular surface

So I start off with finding critical points (point where all partial derivatives disappear at the same time)

I get the following points:

so that means that all point except the above ones are regular points?

the second theorem gives that is a regular surface

Is this solution correct??

Thank you in advance

Re: regular surface- solution check

I didn't check your detailed computation but yes the argument is correct.

Re: regular surface- solution check

thank you for your reply.

I would appreciate if you can have a look at my other post where I am trying to show that a function is smooth

http://mathhelpforum.com/calculus/21...-function.html