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Chapter 16

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 variables) that give a zero gradient. On a surface this will make the tangent plane horizontal.

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Chapter 16

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  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. Stationary points for satisfy the equations: and and Stationary points for satisfy equations: and Neither Cupped Up 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. Cupped Down Saddle Point Local Min Local Max z z z Critical Values for 1 Variable For one variable functions a local max or min could be determined by looking at the sign of the second derivative. For a surface there are 3 different second derivatives. How do they combine to tell you if it is cupped up or down? y y y x x x Concave Down is negative Concave Up is positive Neither is neither

  3. The Discriminate For a function the discriminate is a combination of all second derivatives that determine if the function is cupped up, down or neither. It is abbreviated with . If the point you are evaluating this at is a stationary point it determines if it is a local max, min or saddle point. • For a surface : • If D is negative it is neither cupped up or down. • If D is positive: • a. If is positive it is cupped up. • b. If is negative it is cupped down. • 3. If D is zero can not tell might be up down or neither. Example Find the critical (stationary) points of the surface to the right and classify them. D Type Point Set each derivative to zero and solve. saddle 0 12 -12 -144 and and Local Min 0 12 12 144 Local Max -12 -12 0 144 Since these two equations are independent the stationary points are all the combinations of x and y. -12 12 0 saddle -144

  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. Point D Type 0 Saddle -16 -4 0 4 -16 Saddle Local Min 0 8 Points: Points:

  5. Example Classify the critical (stationary) points of the surface: . Set to zero and solve. Type Point D 0 0 ? -2 0 Because can not draw a conclusion! The critical point is: How do you tell what this critical point is? Form and look at this for small values of h and k. For all small values of h and k this expression will always be negative that means that the point is a local max since the function’s value is zero there. If the point is a stationary point and the discriminate , for small values of h and k consider: If is negative then the point is a local max. If is positive then the point is a local min. If is neither then the point is a saddle.

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