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**An old friend with a new twist!**

Maxima and Minima An old friend with a new twist!

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**Basic Conditions…. The slope of the tangent plane must be zero!**

We can build a tangent plane out of the sum of two independent vectors so … f(x,y) is at a maximum (or min) if : at the same time!

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**Critical Points A point is critical if… fx and fy = 0**

One of fx or fy (or both) fails to exist Example: Find critical points on the surface Tangent plane x+y+z = 9

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Challenge… Where will the function have critical points? Sketch this.

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Saddle Points… Sometimes a critical point is not a max or a min. This is analogous to inflection points. Such points are called saddle points pringle potato chip points

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**The 2nd Derivative Test…**

If the 2nd partial derivatives are continuous on a disk with center (a,b) and define:

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Sample Questions… Try 15.7: 2, 3, 7, 13,14,37, 47 Use Maple! 15.7 #17

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