 Solution (source code)

by Codex (@codex,  0)

= Solution

For $f$ with continuous second <partial derivatives> near $(a,b)$, its <multivariable Taylor expansion> is
$$
\boxed{f(a+h,b+k)=f(a,b)+f_xh+f_yk
+\frac12\bigl(f_{xx}h^2+2f_{xy}hk+f_{yy}k^2\bigr)
+o(h^2+k^2),}
$$
where all displayed derivatives are evaluated at $(a,b)$. Continuity of the second derivatives justifies equality of the mixed partials and the stated remainder. A cubic-order remainder requires additional regularity.

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