My question is how am I supposed to determine what function that is? I think I could handle the partial derivatives after that..
I couldn't find an example showing how to identify this graph anywhere. maybe my graph identification skills aren't that great. I don't know what else to try..
This is not the intention of the exercise. They do not want you to indentify the graph. To explain how they want you to do it, I will do 5(a).
It asks to find $f_x(1,2)$, i.e. how $f$ changes as you move in the $x$ direction from the point $(1,2)$. If you go in the positive $x$ direction from the pink spot which represents $(1,2)$, then you can see that the $z$ co-ordinate (i.e. $f$ value) increases. So the sign of $f_x$ is positive at this point.
Similarly you can do 5(b), 6(a) and 6(b).
When it asks to find $f_{xx}(-1,2)$, this means "the rate at which $f_x$ is changing as you advance in the $x$ direction at this point". At that point, $f_x$ is negative, since the graph is going downwards as you advance in the positive $x$ direction. But $f_x$ is increasing, since the "negative slope" is becoming less steep, like going from gradient of $-2$ to $-1$. So $f_{xx}$ is positive.
Similarly you can do 7(b).
When the two letters are different, you need to find "the rate at which $f_x$ changes as you move in the positive $y$ direction". This, combined with the method for question 7, is how to do 8(a) and 8(b).