1. The problem asks for the second-order Taylor expansion of a function $f(x,y)$ about the point $(a,b)$. 2. The general formula for the second-order Taylor expansion of a function of two variables is: $$ \begin{aligned} f(x,y) \approx & f(a,b) + f_x(a,b)(x - a) + f_y(a,b)(y - b) \\ & + \frac{1}{2} \left[ f_{xx}(a,b)(x - a)^2 + 2f_{xy}(a,b)(x - a)(y - b) + f_{yy}(a,b)(y - b)^2 \right] \end{aligned} $$ 3. Here, $f_x$, $f_y$ are the first partial derivatives, and $f_{xx}$, $f_{xy}$, $f_{yy}$ are the second partial derivatives evaluated at $(a,b)$. 4. Looking at the options: - Option A only includes $f(a,b)$ and the $f_{xx}$ term, missing first derivatives and other second derivatives. - Option B includes only first derivatives but no constant term or second derivatives. - Option C includes constant and first derivatives but no second derivatives. - Option D matches the full second-order Taylor expansion formula except it is truncated but clearly includes the correct terms. 5. Therefore, the correct answer is option D: $$ f(a,b) + f_x(a,b)(x - a) + f_y(a,b)(y - b) + \frac{1}{2} \left[ f_{xx}(a,b)(x - a)^2 + 2f_{xy}(a,b)(x - a)(y - b) + f_{yy}(a,b)(y - b)^2 \right] $$ This formula approximates $f(x,y)$ near $(a,b)$ using values and derivatives at $(a,b)$ up to second order.