The figure below shows a NORMALIZED gradient vector field for the function f(x,

Question

The figure below shows a NORMALIZED gradient vector field for the function f(x, y) = 4 +x^3 + y^3 - 3 x y. Normalized simply means that all non-zero vectors have been given the same length so you can see them better. Use the figure to list all the critical points of f and state whether each is a local minimum, local maximum or saddle point. Use MIN: MAX or SADDLE. NOTE: in order for Web Work to check your answer you need to enter the critical points in what is called dictionary order. This means that point (x, y) comes before (z, w) if x

Explanation / Answer

First critical point is at (0,0) and is SADDLE

Second critical point is at (1,1) and is MAX

Why?

Coming along x=y the (0,0) is stable [inward arrow] but along other eigenvector [perpendicular to it], it is unstable [arrows outward]. Thus a SADDLE

For (1,1) its always outward so it should be MAX.

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