Higher Order Partial Derivatives — Question 5

Question 5 – Solution

Strategy. Integrate $f_{xx}$ and use the other entries to determine the unknown functions of integration.

Step 1: Compatibility The off-diagonal entries agree, and $$\partial_y f_{xx}=-2=\partial_x f_{xy},\qquad \partial_y f_{xy}=4=\partial_x f_{yy}.$$

Step 2: Reconstruct $$f_x=3x^2-2xy+A(y).$$ Since $f_{xy}=-2x+A'(y)=-2x+4y$, $A(y)=2y^2+C$. Integrating in $x$, $$f=x^3-x^2y+2xy^2+Cx+B(y).$$ The condition $f_{yy}=4x+B''(y)=4x+6y$ gives $B=y^3+Dy+E$.

Result $$\boxed{f=x^3-x^2y+2xy^2+y^3+Cx+Dy+E}.$$ Conversely, direct differentiation shows every choice of $C,D,E$ has exactly the proposed Hessian.