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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?

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  • $\begingroup$ 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. $\endgroup$
    – user314
    Jun 21 '20 at 18:51
  • $\begingroup$ 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? $\endgroup$
    – user314
    Jun 21 '20 at 19:07
  • $\begingroup$ 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? $\endgroup$
    – user314
    Jun 22 '20 at 0:54
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$f_{xy}$ 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).

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  • $\begingroup$ 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 $\endgroup$
    – user314
    Jun 22 '20 at 0:37
  • $\begingroup$ 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 $\endgroup$
    – user314
    Jun 22 '20 at 0:51
  • $\begingroup$ Wait nvm, its incorrect. I was only considering the x direction $\endgroup$
    – user314
    Jun 22 '20 at 0:52
  • $\begingroup$ Its a bit tricky to visualise. Look only at the grid lines that go from right to left, pick the one that passes through the points of interest (call it L2), and the ones before (L1) and after (L3) in the y direction. Let's consider point (1,2) - you can see that at that point the slope of the grid line L3 is higher than that of L2 , which in turn is higher than that of L1. Does that help? $\endgroup$ Jun 22 '20 at 1:43
  • $\begingroup$ 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. $\endgroup$
    – user314
    Jun 22 '20 at 2:46

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