Maxima and Minima

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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 xyz 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