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Unformatted text preview: 764 C H A P T E R 15 DIFFERENTIATION IN SEVERAL VARIABLES (ET CHAPTER 14) We thus showed that if the Fermat point exists, then A < 120 ◦ . Similarly, one shows also that B and C must be smaller than 120 ◦ . We conclude that if one of the angles in ” ABC is equal or greater than 120 ◦ , then the Fermat point does not exist. In that case, the minimum value of f ( x , y ) occurs at a point where f x or f y do not exist, that is, at one of the points A , B , or C . 15.8 Lagrange Multipliers: Optimizing with a Constraint (ET Section 14.8) Preliminary Questions 1. Suppose that the maximum of f ( x , y ) subject to the constraint g ( x , y ) = 0 occurs at a point P = ( a , b ) such that ∇ f P ”= 0. Which of the following are true? (a) ∇ f P is tangent to g ( x , y ) = 0 at P . (b) ∇ f P is orthogonal to g ( x , y ) = 0 at P . SOLUTION (a) Since the maximum of f subject to the constraint occurs at P , it follows by Theorem 1 that ∇ f P and ∇ g P are parallel vectors. The gradient ∇ g P is orthogonal to g ( x , y ) = 0 at P , hence ∇ f P is also orthogonal to this curve at P . We conclude that statement (b) is false (yet the statement can be true if ∇ f P = ( , ) ). (b) This statement is true by the reasoning given in the previous part. 2. Figure 8 shows a constraint g ( x , y ) = 0 and the level curves of a function f . In each case, determine whether f has a local minimum, local maximum, or neither at the labeled point. 4 3 2 1 1 2 3 4 A B g ( x , y ) = g ( x , y ) = ∇ f ∇ f FIGURE 8 SOLUTION The level curve f ( x , y ) = 2 is tangent to the constraint curve at the point A . A close level curve that intersects the constraint curve is f ( x , y ) = 1, hence we may assume that f has a local maximum 2 under the constraint at A . The level curve f ( x , y ) = 3 is tangent to the constraint curve. However, in approaching B under the constraint, from one side f is increasing and from the other side f is decreasing. Therefore, f ( B ) is neither local minimum nor local maximum of f under the constraint. 3. On the contour map in Figure 9: (a) Identify the points where ∇ f = λ ∇ g for some scalar λ . (b) Identify the minimum and maximum values of f ( x , y ) subject to g ( x , y ) = 0. 2 6- 2 2 6 Graph of g ( x , y ) = Contour plot of f ( x , y ) (contour interval 2)- 2- 4- 6 y x FIGURE 9 Contour map of f ( x , y ) ; contour interval 2. SOLUTION (a) The gradient ∇ g is orthogonal to the constraint curve g ( x , y ) = 0, and ∇ f is orthogonal to the level curves of f . These two vectors are parallel at the points where the level curve of f is tangent to the constraint curve. These are the points A , B , C , D , E in the figure: S E C T I O N 15.8 Lagrange Multipliers: Optimizing with a Constraint (ET Section 14.8) 765 2 6- 2 2 6 g ( x , y ) = ∇ f A , ∇ g A A E C D B- 2- 6- 6 (b) The minimum and maximum occur where the level curve of f is tangent to the constraint curve. The level curves tangent to the constraint curve are...
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This note was uploaded on 07/15/2009 for the course MATH 210 taught by Professor Hubscher during the Spring '08 term at University of Illinois at Urbana–Champaign.
- Spring '08