Multi-Variable Calculus: Partial Derivatives Using Level Curves

[Dembadon]

Homework Statement

This is a bonus problem on our homework, and I'm having trouble figuring out how to setup what I need.

Homework Equations

Here are my best guesses:
$$f_x=\frac{\partial f}{\partial x}$$
$$f_y=\frac{\partial f}{\partial y}$$
$$f_{xx}=\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right)$$
$$f_{xy}=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)$$
$$f_{yy}=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right)$$
$$f_x=\frac{f(x_0+h,y_0)-f(x,y)}{h}$$
$$f_y=\frac{f(x_0,y_0+h)-f(x,y)}{h}$$

The Attempt at a Solution

My professor said that a "sufficiently precise qualitative explanation" (whatever the hell that means) will be good enough. If it's possible, I would rather provide an analytical explanation. Maybe with the limit definition of the derivative? I'm having trouble figuring out what I need to use, and I have a feeling it's embarrassingly simple.

Initial observations:

(a) If $f$ is increasing at $P$ then $\frac{\partial f}{\partial x}$ is positive. If $f$ is decreasing, then $\frac{\partial f}{\partial x}$ is negative, right?

(b) Same line of reasoning from (a), but holding $x$ constant.

(c) For $f_{xx}$ the contours appear to be closer together for $x<x_0$ than for $x>x_0$. This indicates that $f_{xx}$ is negative, right?

(d) For $f_{xy}$ , I think this means that I'm supposed to observe how $f_x$ changes when $f_y$ changes, right?

(e) For $f_{yy}$ the contours appear to be closer together for $y>y_0$ than for $y<y_0$. This indicates that $f_{yy}$ is positive, right?

 

[CompuChip]

Intuitively, f_x is something like "the (instantaneous) change in f in the x-direction only" - so what do you expect?

You can confirm this suspicion analytically if you look at the definition of f_{x[/b]. Just take h > 0 and check the signs of the denominator and the numerator (you'll have to assume that the level curves are a reasonable representation, e.g. that f doesn't rise to 12 in between the level curves for 6 and 4).}