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Assuming I have a point on a level curves graph for function f(x,y), how would I determine whether the first and second partial derivatives are positive, negative, or zero? I understand that for a regular graph, the slope and concavity would be the indicator, but how would it work with level curves? (partial derivatives with respect to x, y, xx, yy, and xy)enter image description here

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Hint: The gradient $\nabla g$ at a point $(x_0,y_0)$ on a level curve $g(x,y)=k$ is always normal to the curve at $(x_0,y_0)$. You can use the direction of the normal vector to work out the signs of $f_x, f_y$ at $(x_0,y_0)$.

Alternatively, you could ask "How does $f$ change if I move to the right?" If the level curves decrease in number moving right from the given point, then $f_x$ is negative etc. You can repeat this by moving "up" to find the sign of $f_y$.

The obtain the signs of the $2nd$ partial derivatives, consider recreating the shape of the surface from the level curves. A good way to do this is to look at the distances between subsequent level curves and ask if the curve if increasing (going up) or decreasing (going down).

You can repeat the technique of moving to the "right" with $f_x$ as your function to find the sign of $f_{xx}$, moving "up" with $f_x$ to find the sign of $f_{xy}$ etc.

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