Partial Derivatives: Proving & Evaluating at (0,0)

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SANGHERA.JAS
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Do I need to use Schwarz's or Young's theorems to prove it, if don't then do I need to evaluate them on (0,0) using definition.
 

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I would start by taking the mixed partials (fxy and fxy) for each of the given functions, and evaluating these partials at the origin.