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Question : Find all points (x, y) where f(x, y) = e^(x^2 + y^2) has a possible relative maximum or minimum : 2163362

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

18) Find all points (x, y) where f(x, y) = e^{(x^2 + y^2}) has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

A) (0, 1) gives a relative maximum point

B) (0, 1) gives a relative minimum point

C) (0, 0) gives a relative maximum point

D) (0, 0) gives a relative minimum point

19) Let f(x, y) = x^{2} - xy + y^{2} + 2y - 4. The point (- (2/3), - (4/3)) is a

A) not a relative extreme point

B) relative minimum

C) absolute maximum

D) relative maximum

20) Let Q(x, y) = x^{2}y + y^{3}x^{4}. The point (1, 0) is

A) absolute maximum

B) a relative maximum point

C) a relative minimum point

D) not a relative extreme point

21) The function H(x, y) = x^{4} - 9y^{2} - 2x^{2}y + 20y + 4 has

A) a relative minimum at the point (0, 0)

B) neither a relative maximum nor minimum at (1, 4)

C) a relative maximum at the point (0, 1)

D) a relative maximum at the point (-1, 1)

E) none of these

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

22) Let f(x, y) = xe^{2y} + ye^{2x}. Compute (∂^{2}f/∂x^{2}).

Enter your answer as just an unlabeled polynomial in e^{2x} in standard form.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

23) A tennis racket manufacturer produces two types of rackets, standard and competition. The weekly revenue function, in dollars, for x standard rackets and y competition rackets is given by

R(x, y) = 54x + 2xy + 398y - 2x^{2} - 9y^{2}

i) How many of each type of racket must be produced each week to maximize revenue?

ii) What is the maximum weekly revenue?

A) i) 25 standard rackets and 26 competition rackets;

ii) $5664

B) i) 26 standard rackets and 26 competition rackets;

ii) $5668

C) i) 25 standard rackets and 25 competition rackets;

ii) $5675

D) i) 26 standard rackets and 25 competition rackets;

ii) $5677

24) A rectangular box of length x, width y, and height z with no top is to be constructed having a volume of 32 cubic inches. Determine the dimensions that will require the least amount of material to construct the box.

A) 2 inches by 2 inches by 4 inches

B) 4 inches by 4 inches by 2 inches

C) 4 inches by 2 inches by 4 inches

D) 4 inches by 4 inches by 4 inches

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

25) Find the greatest possible volume of a rectangular box that has length plus girth equal to 60 inches. Enter your answer as a single integer (no units).

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Solve the problem.

26) Suppose that the labor cost for a building is approximated by C(x,y) = 2x^{2} + 3y^{2} - 400x - 420y + 20,000, where x is the number of days of skilled labor and y is the number of days of semiskilled labor required. Find the x and y that minimize cost C.

A) x = 100, y = 70

B) x = 200, y = 140

C) x = 70, y = 210

D) x = 210, y = 210

27) The profit (in thousands of dollars) that a company earns from producing x tons of

brass and y tons of steel can be approximated by P(x, y) = 36xy - 8x^{3} - y^{3}. Find the amount of brass and steel that maximize profit and find the value of the maximum profit.

A) (18/5) tons of brass and (9/2) tons of steel; maximum profit is $118,833

B) 6 tons of brass and 3 tons of steel; maximum profit is $216,000

C) (18/7) tons of brass and (36/5) tons of steel; maximum profit is $157,977

D) 3 tons of brass and 6 tons of steel; maximum profit is $216,000

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

28) Let f(x, y) = 8x - 2y. Find the point(s) where f(x, y) may have a possible relative maximum or minimum, subject to the constraint x^{2} + (1/2)y = 18. Use the method of Lagrange multipliers.

Enter your answer as just (a, b) where a, b are integers.

29) Let f(x, y) = x^{2} - 6xy + 10. Find the point(s) where f(x, y) may have a possible relative maximum or minimum, subject to the constraint that 5x + 3y = 11. Use the method of Lagrange multipliers.

Enter your answer as just (a, b) where a, b are both integers.

30) Let f(x, y, z) = xyz. Find the point(s) where f(x, y, z) may have a possible relative maximum or minimum, subject to the constraint x + 6y + 3z = 36 and where x > 0, y > 0, z > 0. Use the method of Lagrange multipliers.

Enter your answer as just (a, b, c) where a, b, c are all integers.

31) Minimize the function f(x) = x + y, subject to the constraint xy = 100, x > 0, y > 0. Use the method of Lagrange multipliers.

Enter your answer exactly as just (a, b), c where (a, b) gives the minimum and c is the Lagrange multiplier as a reduced fraction of form (d/e) (no words or labels).

32) Determine the minimum of f(x, y) = x^{2} + 2y^{2} subject to the constraint x - 2y + 3 = 0.

Enter your answer exactly as just a, b where a is the minimum and b is the Lagrange multiplier, both as integers (no labels).

33) Determine the maximum value of f(x, y) = 4 - x^{2} - y^{2} subject to the constraint y = 3x - 4 .

Enter your answer as exactly just a, b where a is the maximum and b is the Lagrange multiplier as either integers or reduced fractions of form (c/d) (no words or labels).

34) Determine the minimum value of f(x, y) = x^{2} - xy + 2y^{2} + 4 subject to the constraint x - y - 1 = 0.

Enter your answer exactly as just a, b where a is the minimum and b is the Lagrange multiplier, using integers or reduced fractions of form (c/d) (no words or units).