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Consider the function \[f(x,y) = xy - \dfrac{1}{9} (x^3+y^3).\] The critical points (\((0,0)\) and \((3,3)\)) occur when \(\nabla f = \langle f_x, f_y \rangle = \langle 0, 0 \rangle\). There are no other critical points since \(f(x,y)\) is differentiable for all \(x\) and \(y\).
The diagram shows the surface of \(f(x,y)\) and its level curve containing the point \(P(x,y)\). As \(P\) approach the critical points, \(\nabla f \to \langle 0,0 \rangle\)
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Joseph Lo