## Critical Points of a differentiable function

Scroll down to the bottom to view the interactive graph.

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, $$\nabla f \to \langle 0,0 \rangle$$

To view the interactive graph:

1. Make sure you have the latest version of Java 7 installed in your computer. Tablets and smartphones are not supported.