 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/10/solution

f_{x} = 2 x y^{3}

and

f_{y} = 3 x^{2} y^{2}

. Differentiating again gives the four second-order partial derivatives:

f_{xx} = 2 y^{3}, f_{x y} = 6 x y^{2}, f_{y x} = 6 x y^{2}, f_{yy} = 6 x^{2} y .

The two mixed partial derivatives agree, as expected for this smooth polynomial.

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