Question: Consider

1. Compute \(f_{x}\) and \(f_{y}\). Are the first partial derivatives continuous?
2. Show that the mixed second partial derivatives at the origin are not equal, i.e, \(f_{x y} \neq f_{y x}\).
3. In reference to Clairaut's Theorem, explain why this is allowed to happen. In other words, why does that theorem fail to apply?
4. Using Fubini's theorem, we can prove Clairaut's Theorem. To do so, first argue that if \(f_{x y}\) and \(f_{y x}\) are continuous at a point \((a, b)\) and not equal, then one of them is greater than the other in a square \(S\) about that point.
5. Now compute
   \[\iint_{S}\left(\frac{\partial^{2} f}{\partial x \partial y}(x, y)-\frac{\partial^{2} f}{\partial y \partial x}(x, y)\right) d A\]
   using Fubini's theorem. Why does this give you a contradiction? How does this prove Clairaut's theorem?
6. Explain how this proof fails for the given function \(f\).

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