Partial Derivative Signs Through Level Curves

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Homework Statement



Question 2 from http://math.berkeley.edu/~mcivor/math53su11/solutions/hw6solution.pdf here.

I do not understand b) and e). How do I think of the slope with respect to y?

Homework Equations





The Attempt at a Solution



I do know that the partial derivatives are the slopes of the tanget lines. I do know that fx < 0 as the slope is negative there (at point P), but I do not understand how the slop in terms of y looks.
 

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Office_Shredder
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When you say "the slope", it's not clear what you mean because there is a slope in every direction from P for this function. If you did the correct procedure to figure out the sign of fx you should be able to do the exact same thing to figure out fy - why don't you elaborate on how you did fx?
 
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When you say "the slope", it's not clear what you mean because there is a slope in every direction from P for this function. If you did the correct procedure to figure out the sign of fx you should be able to do the exact same thing to figure out fy - why don't you elaborate on how you did fx?
Well, the way I solved for fx was as follows: draw a cross section of the function's 2D hill image (similar to y=1/x). Then place my point p. As the line tangent through P has a negative slope, fx is negative. Actually, I would appreciate help with getting fxx as well - I don't understand how "b/c the x-slope becomes less negative as we move right" correlates with fxx. If fx is the tangent to the curve, then what is fxx geometrically? Answering that would help me a lot.

EDIT: Just googled the second derivative. Now I remember that it tells us the concavity of the quadratic. In this case, is that why fxx is positive? Is that the entire solution to part c) ?

Thanks.
 
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