An old friend with a new twist!

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Presentation transcript:

An old friend with a new twist! Maxima and Minima An old friend with a new twist!

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!

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

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

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

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

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