Word Count : 385

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For part (a), we check if the function satisfies Schwarz's theorem (equality of mixed partial derivatives) at ((1,1)). Let the function be (f(x,y)). Compute the second-order partial derivatives: [ f_{xy} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right), \quad f_{yx} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) ] Evaluate both at ((1,1)). If (f_{xy}(1,1) = f_{yx}(1,1)), the theorem is satisfied. If not, it fails. Without the explicit function given in your query, the method is: compute both mixed partials and compare their _____ ____ _______ ___ _________ _________ ___ ___.
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