[Math] Partial Derivatives by Level Curves

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The level curves of a function f(x,y) is given in the image below.

enter image description here

I wish to determine the sign of the partial derivatives:

$f_{x} , f_{y} , f_{xx} , f_{yy} , f_{xy} , f_{yx}$

at the point P given in the image.

I am not sure how to solve this problem, never seen anything similar.

I imagine that the solution goes round the fact that the level curves values are reducing when x increases, so it must say something about the shape of the function and the slope that is being created, but I can't see how x or y creates the plane here, and definitely can't think of the second order derivatives.

Your help will be most appreciated, I find this problem interesting, as it's not technical, it demands understanding.

Best Answer

HINT

Observe that $P$ is on the curvee level $f(x,y)=6$ and $f(x,y)$ decreases for $x$ increasing thus

  • $f_x=\frac{\partial f}{\partial x}<0$

note also that from $P$ level curves are less dense with $x$ increasing thus

  • $f_{xx}=\frac{\partial^2 f}{\partial x^2}>0$
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