1. **Problem Statement:** Find the second-order Taylor expansion of the function $f(x,y)$ about the point $(a,b)$. 2. **Formula for the second-order Taylor expansion:** $$ T_2(x,y) = f(a,b) + f_x(a,b)(x - a) + f_y(a,b)(y - b) + \frac{1}{2} \left[ f_{xx}(a,b)(x - a)^2 + 2 f_{xy}(a,b)(x - a)(y - b) + f_{yy}(a,b)(y - b)^2 \right] $$ 3. **Explanation:** - $f(a,b)$ is the value of the function at the point $(a,b)$. - $f_x(a,b)$ and $f_y(a,b)$ are the first partial derivatives with respect to $x$ and $y$ at $(a,b)$. - $f_{xx}(a,b)$, $f_{xy}(a,b)$, and $f_{yy}(a,b)$ are the second partial derivatives at $(a,b)$. - The terms $(x - a)$ and $(y - b)$ represent the displacement from the point $(a,b)$. 4. **Interpretation of options:** - Option A includes only $f(a,b)$ and a second derivative term in $x$. - Option B includes only first derivatives terms. - Option C includes $f(a,b)$ and first derivatives terms. - Option D includes $f(a,b)$, first derivatives, and all second derivative terms with the correct coefficients. 5. **Conclusion:** The correct second-order Taylor expansion is given by option D. **Final answer:** $$ \boxed{\text{D: } f(a,b) + f_x(a,b)(x - a) + f_y(a,b)(y - b) + \frac{1}{2} \left[ f_{xx}(a,b)(x - a)^2 + 2 f_{xy}(a,b)(x - a)(y - b) + f_{yy}(a,b)(y - b)^2 \right]} $$