Answer
See proof
Work Step by Step
Step 1
To determine the sign of the partial derivative $\frac{\partial f}{\partial x}$ from the graph at a point $(x_0, y_0)$, we need to know how the $z$ coordinate (value of $f$) changes when you move in the positive $x$ direction from point $(x_0, y_0)$: $\rightarrow$ If $z$ increases, then $\frac{\partial f}{\partial x}(x_0, y_0)$ is positive. $\rightarrow$ If $z$ decreases, then $\frac{\partial f}{\partial x}(x_0, y_0)$ is negative.
Step 2 To determine the sign of the partial derivative $\frac{\partial f}{\partial y}$ from the graph at a point $(x_0, y_0)$, we need to know how the $z$ coordinate (value of $f$) changes when you move in the positive $y$ direction from point $(x_0, y_0)$: $\rightarrow$ If $z$ increases, then $\frac{\partial f}{\partial y}(x_0, y_0)$ is positive. $\rightarrow$ If $z$ decreases, then $\frac{\partial f}{\partial y}(x_0, y_0)$ is negative.
