Explanation Given Information: The provided function is z = x 2 + y 2 − 4 . Formula used: Differentiate z = f ( x , y ) with respect to x by holding y constant as: ∂ ∂ x [ f ( x , y ) ] = f x ( x , y ) Differentiate f ( x , y ) with respect to y by holding x constant as: ∂ ∂ y [ f ( x , y ) ] = f y ( x , y ) The every differentiable function f ( x , y ) , ∂ 2 ∂ y ∂ x f ( x , y ) = ∂ 2 ∂ x ∂ y f ( x , y ) The following procedure are used to calculate relative maximum and saddle point of given function f ( x , y ) . Step-1: Calculate first derivative of function f ( x , y ) with respect to x and y . Step-2 Calculate second derivative of function f ( x , y ) with respect to x and y . Step-3: Equate first derivative of function f ( x , y ) with respect to x to zero. Step-4: Equate first derivative of function f ( x , y ) with respect to y to zero. Step-5: Calculate critical point of function f ( x , y ) that is ( a , b ) . Step-6: Now test function f ( x , y ) at critical point ( a , b ) for extrema and saddle point that is describe in following table. Critical point ∂ 2 ∂ 2 x f ( x , y ) ∂ 2 ∂ x 2 f ( x , y ) ∂ 2 ∂ y 2 f ( x , y ) − [ ∂ 2 ∂ y ∂ x f ( x , y ) ] 2 Conclusion ( a , b ) ∂ 2 ∂ 2 x f ( a , b ) > 0 ∂ 2 ∂ x 2 f ( a , b ) ∂ 2 ∂ y 2 f ( a , b ) − [ ∂ 2 ∂ y ∂ x f ( a , b ) ] 2 > 0 The function f ( x , y ) has relative minimum at point ( a , b ) ( a , b ) ∂ 2 ∂ 2 x f ( a , b ) < 0 ∂ 2 ∂ x 2 f ( a , b ) ∂ 2 ∂ y 2 f ( a , b ) − [ ∂ 2 ∂ y ∂ x f ( a , b ) ] 2 > 0 The function f ( x , y ) has relative maximum at point ( a , b ) ( a , b ) ∂ 2 ∂ x 2 f ( a , b ) ∂ 2 ∂ y 2 f ( a , b ) − [ ∂ 2 ∂ y ∂ x f ( a , b ) ] 2 < 0 The function f ( x , y ) has saddle point ( a , b , f ( a , b ) ) ( a , b ) ∂ 2 ∂ x 2 f ( a , b ) ∂ 2 ∂ y 2 f ( a , b ) − [ ∂ 2 ∂ y ∂ x f ( a , b ) ] 2 = 0 The test gives no information. The simple power rule of derivative: d d x ( x n ) = n x n − 1 The constant rule of derivative: d d x ( c ) = 0 The derivative of constant coefficient: d d x ( c f ( x ) ) = c f ′ ( x ) Calculation: Consider the function, z = x 2 + y 2 − 4

Mathematical Applications for the Management, Life, and Social Sciences

11th Edition

ISBN: 9781305108042

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

See similar textbooks

Question

Chapter 14.4, Problem 4E

To determine

To calculate: The relative maxima, relative minima, and saddle points of z=x2+y2−4, if they exist.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.4, Problem 3E

Chapter 14.4, Problem 5E

Students have asked these similar questions

pleasd dont use chat gpt

1. True or false: (a) if E is a subspace of V, then dim(E) + dim(E+) = dim(V) (b) Let {i, n} be a basis of the vector space V, where vi,..., are all eigen- vectors for both the matrix A and the matrix B. Then, any eigenvector of A is an eigenvector of B. Justify. 2. Apply Gram-Schmidt orthogonalization to the system of vectors {(1, 2, -2), (1, −1, 4), (2, 1, 1)}. 3. Suppose P is the orthogonal projection onto a subspace E, and Q is the orthogonal projection onto the orthogonal complement E. (a) The combinations of projections P+Q and PQ correspond to well-known oper- ators. What are they? Justify your answer. (b) Show that P - Q is its own inverse. 4. Show that the Frobenius product on n x n-matrices, (A, B) = = Tr(B*A), is an inner product, where B* denotes the Hermitian adjoint of B. 5. Show that if A and B are two n x n-matrices for which {1,..., n} is a basis of eigen- vectors (for both A and B), then AB = BA. Remark: It is also true that if AB = BA, then there exists a common…

Question 1. Let f: XY and g: Y Z be two functions. Prove that (1) if go f is injective, then f is injective; (2) if go f is surjective, then g is surjective. Question 2. Prove or disprove: (1) The set X = {k € Z} is countable. (2) The set X = {k EZ,nЄN} is countable. (3) The set X = R\Q = {x ER2 countable. Q} (the set of all irrational numbers) is (4) The set X = {p.√2pQ} is countable. (5) The interval X = [0,1] is countable. Question 3. Let X = {f|f: N→ N}, the set of all functions from N to N. Prove that X is uncountable. Extra practice (not to be submitted). Question. Prove the following by induction. (1) For any nЄN, 1+3+5++2n-1 n². (2) For any nЄ N, 1+2+3++ n = n(n+1). Question. Write explicitly a function f: Nx N N which is bijective.

Chapter 14 Solutions

Mathematical Applications for the Management, Life, and Social Sciences

Ch. 14.1 - CHECKPOINT 1. Find the domain of the function Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E Ch. 14.1 - Prob. 6E Ch. 14.1 - Prob. 7E Ch. 14.1 - Prob. 8E Ch. 14.1 - Prob. 9E

Ch. 14.1 - Prob. 10E Ch. 14.1 - Prob. 11E Ch. 14.1 - Prob. 12E Ch. 14.1 - Prob. 13E Ch. 14.1 - Prob. 14E Ch. 14.1 - Prob. 15E Ch. 14.1 - Prob. 16E Ch. 14.1 - In Problems 15-22, evaluate each function as...Ch. 14.1 - Prob. 18E Ch. 14.1 - Prob. 19E Ch. 14.1 - Prob. 20E Ch. 14.1 - Prob. 21E Ch. 14.1 - In Problems 15-22, evaluate each function as...Ch. 14.1 - Prob. 23E Ch. 14.1 - Prob. 24E Ch. 14.1 - Prob. 25E Ch. 14.1 - Temperature-humidity models There are different...Ch. 14.1 - Prob. 28E Ch. 14.1 - 29. Mortgage The following tables show that a...Ch. 14.1 - 30. Wind chill Wind and cold temperatures combine...Ch. 14.1 - Prob. 31E Ch. 14.1 - Prob. 32E Ch. 14.1 - Prob. 33E Ch. 14.1 - Prob. 34E Ch. 14.1 - Prob. 35E Ch. 14.1 - 36. Profit The Kirk Kelly Kandy Company makes two...Ch. 14.1 - Prob. 37E Ch. 14.1 - Prob. 38E Ch. 14.2 - Prob. 1CP Ch. 14.2 - Prob. 2CP Ch. 14.2 - Prob. 3CP Ch. 14.2 - Prob. 4CP Ch. 14.2 - Prob. 5CP Ch. 14.2 - Prob. 1E Ch. 14.2 - Prob. 2E Ch. 14.2 - Prob. 3E Ch. 14.2 - Prob. 4E Ch. 14.2 - Prob. 5E Ch. 14.2 - Prob. 6E Ch. 14.2 - Prob. 7E Ch. 14.2 - Prob. 8E Ch. 14.2 - Prob. 9E Ch. 14.2 - Prob. 10E Ch. 14.2 - Prob. 11E Ch. 14.2 - Prob. 12E Ch. 14.2 - Prob. 13E Ch. 14.2 - Prob. 14E Ch. 14.2 - Prob. 15E Ch. 14.2 - Prob. 16E Ch. 14.2 - Prob. 17E Ch. 14.2 - Prob. 18E Ch. 14.2 - 19. Find the slope of the tangent in the...Ch. 14.2 - 20. Find the slope of the tangent in the...Ch. 14.2 - Prob. 21E Ch. 14.2 - Prob. 22E Ch. 14.2 - Prob. 23E Ch. 14.2 - Prob. 24E Ch. 14.2 - Prob. 25E Ch. 14.2 - Prob. 26E Ch. 14.2 - Prob. 27E Ch. 14.2 - Prob. 28E Ch. 14.2 - Prob. 29E Ch. 14.2 - Prob. 30E Ch. 14.2 - Prob. 31E Ch. 14.2 - 32. If , find the following. (a) (b) (c) (d) Ch. 14.2 - 33. If , find the following. Ch. 14.2 - 34. If , find the following. Ch. 14.2 - Prob. 35E Ch. 14.2 - Ch. 14.2 - Ch. 14.2 - Prob. 38E Ch. 14.2 - Prob. 39E Ch. 14.2 - Prob. 40E Ch. 14.2 - 41. . Ch. 14.2 - . Ch. 14.2 - Prob. 43E Ch. 14.2 - Ch. 14.2 - 45. Mortgage When a homeowner has a 25-year...Ch. 14.2 - 46. Mass transportation ridership Suppose that in...Ch. 14.2 - 47. Wilson's lot size formula In economics, the...Ch. 14.2 - 48. Cost Suppose that the total cost (in dollars)...Ch. 14.2 - 49. Pesticide Suppose that the number of thousands...Ch. 14.2 - 50. Profit Suppose that the profit (in dollars)...Ch. 14.2 - Prob. 51E Ch. 14.2 - Prob. 52E Ch. 14.2 - 53. Production Suppose that the output Q (in...Ch. 14.2 - Prob. 54E Ch. 14.2 - Prob. 55E Ch. 14.2 - Prob. 56E Ch. 14.3 - CHECKPOINT If the joint cost in dollars for two...Ch. 14.3 - Prob. 2CP Ch. 14.3 - Prob. 3CP Ch. 14.3 - Prob. 1E Ch. 14.3 - Prob. 2E Ch. 14.3 - 3. The total cost of producing 1 unit of a product...Ch. 14.3 - Prob. 4E Ch. 14.3 - Prob. 5E Ch. 14.3 - Prob. 6E Ch. 14.3 - Prob. 7E Ch. 14.3 - Prob. 8E Ch. 14.3 - 9. If the joint cost function for two products is ...Ch. 14.3 - 10. Suppose the joint cost function for x units of...Ch. 14.3 - 11. Suppose that the joint cost function for two...Ch. 14.3 - Prob. 12E Ch. 14.3 - Prob. 13E Ch. 14.3 - Prob. 14E Ch. 14.3 - Prob. 15E Ch. 14.3 - Prob. 16E Ch. 14.3 - Prob. 17E Ch. 14.3 - Prob. 18E Ch. 14.3 - Prob. 19E Ch. 14.3 - Prob. 20E Ch. 14.3 - 21. Suppose the Cobb-Douglas production function...Ch. 14.3 - Prob. 22E Ch. 14.3 - Prob. 23E Ch. 14.3 - Prob. 24E Ch. 14.3 - Prob. 25E Ch. 14.3 - Prob. 26E Ch. 14.3 - Prob. 27E Ch. 14.3 - Prob. 28E Ch. 14.3 - Prob. 29E Ch. 14.3 - Prob. 30E Ch. 14.4 - CHECKPOINT Suppose that Find Ch. 14.4 - Prob. 2CP Ch. 14.4 - Prob. 3CP Ch. 14.4 - Prob. 4CP Ch. 14.4 - Prob. 1E Ch. 14.4 - Prob. 2E Ch. 14.4 - Prob. 3E Ch. 14.4 - Prob. 4E Ch. 14.4 - Prob. 5E Ch. 14.4 - Prob. 6E Ch. 14.4 - Prob. 7E Ch. 14.4 - Prob. 8E Ch. 14.4 - Prob. 9E Ch. 14.4 - Prob. 10E Ch. 14.4 - Prob. 11E Ch. 14.4 - Prob. 12E Ch. 14.4 - Prob. 13E Ch. 14.4 - Prob. 14E Ch. 14.4 - Prob. 15E Ch. 14.4 - Prob. 16E Ch. 14.4 - Prob. 17E Ch. 14.4 - Prob. 18E Ch. 14.4 - 19. Profit Suppose that the quarterly profit from...Ch. 14.4 - Prob. 20E Ch. 14.4 - 21. Nutrition A new food is designed to add weight...Ch. 14.4 - Prob. 22E Ch. 14.4 - 23. Production Suppose that tons is the...Ch. 14.4 - 24. Production Suppose that x units of one input...Ch. 14.4 - 25. Profit Suppose that a manufacturer produces...Ch. 14.4 - Prob. 26E Ch. 14.4 - Prob. 27E Ch. 14.4 - Prob. 28E Ch. 14.4 - 29. Profit A company manufactures two products, A...Ch. 14.4 - Prob. 30E Ch. 14.4 - Prob. 31E Ch. 14.4 - The manager of the Sea Islands Chicken Shack is...Ch. 14.4 - Prob. 34E Ch. 14.4 - Prob. 35E Ch. 14.4 - Prob. 36E Ch. 14.5 - Prob. 1CP Ch. 14.5 - Prob. 2CP Ch. 14.5 - Prob. 3CP Ch. 14.5 - Prob. 4CP Ch. 14.5 - Prob. 1E Ch. 14.5 - Prob. 2E Ch. 14.5 - Prob. 3E Ch. 14.5 - Prob. 4E Ch. 14.5 - Prob. 5E Ch. 14.5 - Prob. 6E Ch. 14.5 - Prob. 7E Ch. 14.5 - Prob. 8E Ch. 14.5 - Prob. 9E Ch. 14.5 - Prob. 10E Ch. 14.5 - Prob. 11E Ch. 14.5 - Prob. 12E Ch. 14.5 - Prob. 13E Ch. 14.5 - Prob. 14E Ch. 14.5 - Prob. 15E Ch. 14.5 - 16. Utility Suppose that the budget constraint in...Ch. 14.5 - 17. Utility Suppose that the utility function for...Ch. 14.5 - 18. Utility Suppose that the utility function for...Ch. 14.5 - Prob. 19E Ch. 14.5 - Prob. 20E Ch. 14.5 - 21. Cost A firm has two plants, X and Y. Suppose...Ch. 14.5 - Prob. 22E Ch. 14.5 - Prob. 23E Ch. 14.5 - Prob. 24E Ch. 14.5 - 25. Manufacturing Find the dimensions (in...Ch. 14.5 - Prob. 26E Ch. 14 - 1. What is the domain of ? Ch. 14 - Prob. 2RE Ch. 14 - Prob. 3RE Ch. 14 - 4. If . Ch. 14 - Prob. 5RE Ch. 14 - Prob. 6RE Ch. 14 - Prob. 7RE Ch. 14 - Prob. 8RE Ch. 14 - Prob. 9RE Ch. 14 - Prob. 10RE Ch. 14 - Prob. 11RE Ch. 14 - Prob. 12RE Ch. 14 - Prob. 13RE Ch. 14 - Prob. 14RE Ch. 14 - In Problems 15-18, find the second partials. Ch. 14 - Prob. 16RE Ch. 14 - Prob. 17RE Ch. 14 - Prob. 18RE Ch. 14 - Prob. 19RE Ch. 14 - Prob. 20RE Ch. 14 - Prob. 21RE Ch. 14 - Prob. 22RE Ch. 14 - Prob. 23RE Ch. 14 - Prob. 24RE Ch. 14 - Prob. 25RE Ch. 14 - Prob. 26RE Ch. 14 - Prob. 27RE Ch. 14 - Prob. 28RE Ch. 14 - Prob. 29RE Ch. 14 - Prob. 30RE Ch. 14 - Prob. 31RE Ch. 14 - Prob. 32RE Ch. 14 - Prob. 33RE Ch. 14 - Prob. 34RE Ch. 14 - 35. Modeling US. average wage The table gives the...Ch. 14 - Prob. 36RE Ch. 14 - Prob. 1T Ch. 14 - Prob. 2T Ch. 14 - Prob. 3T Ch. 14 - Prob. 4T Ch. 14 - Prob. 5T Ch. 14 - Prob. 6T Ch. 14 - 7. Suppose the demand functions for two products...Ch. 14 - Prob. 8T Ch. 14 - 9. Find x and y that maximize the utility function...Ch. 14 - Prob. 10T

Knowledge Booster

The relative maxima , relative minima , and saddle points of z = x 2 + y 2 − 4 , if they exist.

Solution Summary: The author explains how to calculate the relative maxima, relative minima and saddle points of z=x2+y2.

To calculate: The relative maxima, relative minima, and saddle points of z=x2+y2−4, if they exist.

Want to see the full answer?

Chapter 14 Solutions