# How to determine the signs of mixed partial derivatives from a graph?

Graph

My question is 8a) and 8b) from the image above.

• In 8a), the question is asking for the rate of change of $$f_{x}$$ in the y direction (this is what I think). From the graph, $$f_{x}$$ is increasing while x increases, so $$f_{xy}$$ is decreasing as the plane is curving down in the positive y direction, hence the sign of $$f_{xy}(1,2)$$ is negative, but the correct answer is positive.

• For 8b), the sign of $$f_{xy}(-1,2)$$ is negative, but I do not know why that is.

My interpretation is that $$f_{xy}(1,2)$$ and $$f_{xy}(-1,2)$$ should both be negative since both would be the same as finding the sign of $$f_{y}(1,2)$$ and $$f_{y}(-1,2)$$ from 5b) and 6b).

So my question is why are the answers for 8a) and 8b) positive and negative, rather than negative and negative?

• If it was $f_{xx}$ or $f_{yy}$, it would be the second partial derivative dealing with concavity, but I am clueness on what $f_{xy}$ and $f_{yx}$ mean. Jun 21 '20 at 18:51
• Both the points (1,2) and (-1,2) are both moving in the same positive x and y direction denoted by $f_{xy}(1,2)$ and $f_{xy}(-1,2)$, so wouldn't that mean that the rate of change in $f_{x}(1,2)$ and $f_{x}(-1,2)$ in the positive y direction are both negative, as the plane is curving down in the positive y direction? Jun 21 '20 at 19:07
• What does $f_{xy}$ look like geometrically? Is it the rate of change of x while y is constant, then stopping and becoming the rate of change of y with x now being constant? Jun 22 '20 at 0:54

$$f_{xy}$$ is the rate of change of $$f_x$$ in the y direction, so you can't look at the curvature of the surface in the y direction to determine the sign of $$f_{xy}$$. You should look at how the slope in the x direction changes as y changes. Look at the grid lines and verify that as you move in the $$y^+$$ direction near the points (1,2) and (-1,2), $$f_x$$ increases near the former and decreases near the latter (notice that $$|f_x|$$ increases in both cases).
• Since $f_{xy}$ is the rate of change of $f_{x}$ in the positive y direction, the graph shows the grid lines curving down the plane in the positive y direction while moving along the slope $f_{x}$ in the positive x direction ( towards the left of the graph ). What I'm trying to say is if the grid lines curve down (negative) towards the positive y direction as $f_{x}$ changes ( slope increases while moving in the positive x direction) wouldn't $f_{xy}(1,2)$ be negative then? The correct answer is positive but I do not know why Jun 22 '20 at 0:37
• I'm not sure if I've figured out the answer, but if the graph was rotated such that the positive y-axis is coming out towards the viewer, then for $f_{xy}(1,2)$, the slope is of $f_{x}$ while moving in the positive x-direction is increasing from right to left which means the slope is positive? But if viewing the graph as a normal graph the slope would look like a negative slope because it would go from top left to bottom right. I'm not sure Jun 22 '20 at 0:51
• I am not sure what you mean by ($L_{1}$) and ($L_{3}$) in the y direction. You said grid lines that go from right to left which means the x direction, but ($L_{1}$) and ($L_{3}$) are in the y direction. Jun 22 '20 at 2:46