Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Continuous mathematics / 6 / a

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Solution

The derivative is represented by the gradient, and the Hessian matrix contains the second partial derivatives:

\nabla f (x, y) = (4 x^{3} - 4 y 4 y - 4 x), H (f) (x, y) = (12 x^{2} - 4 - 4 4) .

Equivalently the derivative as a linear map is

D f (x, y) (u, v) = (4 x^{3} - 4 y) u + (4 y - 4 x) v

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