1. **Problem statement:** Given the function $$z = x^2 + y^2$$, find the sign of the partial derivatives $$f_x(1,1)$$, $$f_x(-1,1)$$, $$f_y(1,1)$$, and $$f_x(0,0)$$ using the graph of the surface. 2. **Formula for partial derivatives:** The partial derivative with respect to $$x$$ is $$f_x = \frac{\partial}{\partial x}(x^2 + y^2) = 2x$$. The partial derivative with respect to $$y$$ is $$f_y = \frac{\partial}{\partial y}(x^2 + y^2) = 2y$$. 3. **Evaluate each partial derivative at the given points:** - $$f_x(1,1) = 2 \times 1 = 2$$, which is positive. - $$f_x(-1,1) = 2 \times (-1) = -2$$, which is negative. - $$f_y(1,1) = 2 \times 1 = 2$$, which is positive. - $$f_x(0,0) = 2 \times 0 = 0$$, which is zero. 4. **Interpretation:** - A positive partial derivative means the function increases as $$x$$ or $$y$$ increases. - A negative partial derivative means the function decreases as $$x$$ or $$y$$ increases. - Zero means no change at that point in that direction. **Final answers:** - $$f_x(1,1) > 0$$ - $$f_x(-1,1) < 0$$ - $$f_y(1,1) > 0$$ - $$f_x(0,0) = 0$$