 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2013/discrete-mathematics-probability-and-continuous-mathematics/4/a/solution

Take each partial derivative while holding the other coordinate fixed. Differentiating the polynomial term by term gives

f_{x} f_{y} f_{xx} f_{x y} f_{yy} = 2 x (y^{4} + y^{2} + 1), = 4 x^{2} y^{3} + 3 y^{2} + 2 x^{2} y - 3, = 2 (y^{4} + y^{2} + 1), = 8 x y^{3} + 4 x y, = 12 x^{2} y^{2} + 6 y + 2 x^{2} .

Since the function is a polynomial, both orders of the mixed partial derivative agree everywhere.

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