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Chapter 16 Section 16.5 Local Extreme Values. Critical (or Stationary) Points A critical or stationary point is a point (i.e. values for the independent.

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Presentation on theme: "Chapter 16 Section 16.5 Local Extreme Values. Critical (or Stationary) Points A critical or stationary point is a point (i.e. values for the independent."— Presentation transcript:

1 Chapter 16 Section 16.5 Local Extreme Values

2 Critical (or Stationary) Points A critical or stationary point is a point (i.e. values for the independent variables) that give a zero gradient. On a surface this will make the tangent plane horizontal. Extreme Points At a stationary point a surfaced can be cupped up, cupped down or neither, this determines if the point is a local max, local min or saddle point. x y z x y z x y z Cupped Up Cupped Down Neither Local Max Local Min Saddle Point

3 Example Find the critical (stationary) points of the surface to the right and classify them. Set each derivative to zero and solve. Since these two equations are independent the stationary points are all the combinations of x and y. Point D Type saddle Local Min Local Max

4 Example Find the critical (stationary) points of the surface to the right and classify them. Set both equations equal to zero. This system of equations is not independent (i.e. there are x ’s and y ’s in both). We need to solve one and substitute into the other. D Point Type Saddle Local Min

5 Set to zero and solve. D Point Type ?


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