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(b) Use the result you obtained in part (a) to approxi- mate the minimum value of f subject to the constraint (x 4 ) 2 + (y 4 ) 2 = 4. (c) Confirm graphically that you have found a minimum and not a maximum. (d) Check your approximation using Lagrange multipli- ers and solving the required equations numerically. 5–12 Use Lagrange multipliers to find the maximum and min- imum values of f subject to the given constraint. Also, find the points at which these extreme values occur. 5. f(x, y) = xy ; 4 x 2 + 8 y 2 = 16 6. f(x, y) = x 2 y 2 ; x 2 + y 2 = 25 7. f(x, y) = 4 x 3 + y 2 ; 2 x 2 + y 2 = 1 8. f(x, y) = x 3 y 1; x 2 + 3 y 2 = 16 9. f(x, y, z) = 2 x + y 2 z ; x 2 + y 2 + z 2 = 4 10. f(x, y, z) = 3 x + 6 y + 2 z ; 2 x 2 + 4 y 2 + z 2 = 70 11. f(x, y, z) = xyz ; x 2 + y 2 + z 2 = 1 12. f(x, y, z) = x 4 + y 4 + z 4 ; x 2 + y 2 + z 2 = 1 13–16 True–False Determine whether the statement is true or false. Explain your answer. 13. A “Lagrange multiplier” is a special type of gradient vector. 14. The extrema of f(x, y) subject to the constraint g(x, y) = 0 occur at those points for which f = g . 15. In the method of Lagrange mutlipliers it is necessary to solve a constraint equation g(x, y) = 0 for y in terms of x . 16. The extrema of f(x, y) subject to the constraint g(x, y) = 0 occur at those points at which a contour of f is tangent to the constraint curve g(x, y) = 0. 17–24 Solve using Lagrange multipliers. 17. Find the point on the line 2 x 4 y = 3 that is closest to the origin. 18. Find the point on the line y = 2 x + 3 that is closest to ( 4 , 2 ) . 19. Find the point on the plane x + 2 y + z = 1 that is closest to the origin. 20. Find the point on the plane 4 x + 3 y + z = 2 that is closest to ( 1 , 1 , 1 ) . 21. Find the points on the circle x 2 + y 2 = 45 that are closest to and farthest from ( 1 , 2 ) . 22. Find the points on the surface xy z 2 = 1 that are closest to the origin. 23. Find a vector in 3-space whose length is 5 and whose com- ponents have the largest possible sum. 24. Suppose that the temperature at a point (x, y) on a metal plate is T (x, y) = 4 x 2 4 xy + y 2 . An ant, walking on the plate, traverses a circle of radius 5 centered at the origin. What are the highest and lowest temperatures encountered by the ant? 25–32 Use Lagrange multipliers to solve the indicated exer- cises from Section 13.8. 25. Exercise 38 26. Exercise 39
Chapter 13 Review Exercises 997 27. Exercise 40 28. Exercise 41 29. Exercise 43 30. Exercises 45(a) and (b) 31. Exercise 46 32. Exercise 47 33. C Let α , β , and γ be the angles of a triangle. (a) Use Lagrange multipliers to find the maximum value of f(α, β, γ ) = cos α cos β cos γ , and determine the an- gles for which the maximum occurs. (b) Express f(α, β, γ ) as a function of α and β alone, and use a CAS to graph this function of two variables. Con- firm that the result obtained in part (a) is consistent with the graph. 34. The accompanying figure shows the intersection of the el- liptic paraboloid z = x 2 + 4 y 2 and the right circular cyl- inder x 2 + y 2 = 1. Use Lagrange multipliers to find the highest and lowest points on the curve of intersection.