•A surface, concave up at a point in the x and y directions, but with no min or max.

Let x = r cos θ, y = r sin θ, z = r^2 cos (4 θ). Then  ∂ z/∂ x    = 0,  ∂ z/∂ y    = 0,  ∂^2 z/∂ x^2    > 0 and  ∂^2 z/∂ y^2    > 0 at the point (x, y) = (0, 0). (This is a terrific [and subtle] exercise in using the chain rule.) But there is no min or max at (0, 0) .

In[221]:=

g = ParametricPlot3D[(x = Sqrt[rr] Cos[th] ; y = Sqrt[rr] Sin[th] ; {x, y, rr Cos[4 th]}), {rr, 0, 1/3}, {th, 0, 2 Pi}, PlotPoints -> {10, 200}] ;

[Graphics:HTMLFiles/index_9.gif]

Click for the LiveGraphics3D version.

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