If $f(x,y)$ is concave such that $f_1 < 0, f_2 > 0$, how are the level curves supposed to look like? Suppose $f(x,y)$ is a concave function such that $\frac{\partial f}{\partial x} < 0$ and $\frac{\partial f}{\partial y} > 0$.

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*How are the level curves supposed to look like?

*Can I get an example of such a concave function?
Just to be clear, this isn't a homework question. I am learning multivariable calculus and I am stuck with convex and concave functions, especially visualizing the various types of such functions.

My attempt:
To answer (1), I have drawn two possible images below. The curves are essentially $L_c = \{(x,y) : f(x,y) = c\}$ or the level sets of $f$. In each figure, the arrow represents the direction in which the level curves are attaining higher values (of $c$).
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We know that $tf(x) + (1-t)f(y) \leq f(tx + (1-t)y)$ $(t \in (0,1))$ for a concave function. If we pick two points $p$ and $q$ on a level curve, then it's easy to notice that $f(x(r), y(r)) > \max\{f(x(p), y(p)), f(x(q), y(q))\}$. This is only true for the first image and not the second image. Is there a better way than this, just for visualization?
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I don't have an answer to (2).
 A: I agree with your attempt, and I don't see any significantly better way of doing it.
Remember, it's not really possible to arrive at any kind of authoritative answer, given that the pictures give us very little information about the functions (who knows what happens between the level curves, or their precise elevation?). At best, we can use the line segment method to definitively eliminate the second from being concave:

Note how the line segment, between two points of the same higher elevation, crosses the other level curve at a lower elevation. This contradicts the definition of concavity, which means the function is definitely not concave.
Your similar picture gives an example of the concavity definition holding true (which is not enough for a formal proof, but as good as we'll get under the circumstances), but this is an actual counterexample. So, we know the second is not concave, and the first looks to be concave, as best we can tell.
As for an example, I would be lazy and choose:
$$f(x, y) = -x + y,$$
which is linear and hence concave (and convex). Of course, in this case, the level curves are straight lines.
If you want a strictly concave function, you could do
$$f(x, y) = -e^x - e^{-y}.$$
We have $f'_1(x) = -e^x < 0$ and $f'_2(y) = e^{-y} > 0$. Further, $f(x, y)$ is the sum of two concave functions $-e^x$ and $-e^{-y}$, which will make it concave.
