Quick implicit differentiation question

The problem is stated as follows:

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Show that the equation defines z as a function of x and y near (1,1,1). Also show that (1,1) is a stationary point of this function and determine its kind.

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The first part is easily shown with the implicit function theorem.

Differentiating the equation implicitly with respect to x:

, so gives near (x,y) = (1,1), and consequently, there.

Thus, (x,y) = (1,1) isn't a stationary point as the problem claims.

Feel free to point and laugh at me and tell me that (or preferably why) my arithmetic is wrong.