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

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Presentation on theme: "Maxima and Minima An old friend with a new twist!."— Presentation transcript:

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

2 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!

3 Critical Points A point is critical if… f x and f y = 0 One of f x or f y (or both) fails to exist Example: Find critical points on the surface Tangent plane x+y+z = 9

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

5 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

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

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


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