Optimization Problems (11.7-11.8)

15. Let f(x, y) = -²³ - y³ + 3xy.

(a) Find the critical points of f and classify them.

(b) Where are the local maxima and minima of f? What are the maxima and minima?

16. Using the method of Lagrange multipliers set up a system of equations to find the absolute

maximum of ² + y² subject to 6x² + 2y² = 0.

17. Using the method of Lagrange multipliers, find the absolute maximum of f(x, y) = z²+y² subject

to the constraint 2 + y² = 1.

18. Explain in your own words how the method of Lagrange multipliers relates to level curves of a

function of two variables. [Draw some pictures to help in your explanation! If you need a practice

function use f(x, y) = x² - y² subject to the constraint x² + y² = 4.]

Double Integrals in Rectangular Coordinates (12.1-12.3)

19. Find the integral of the function 8³ cos(y) on the region R = [−1, 1] × [-^, ^]

20. Evaluate the integral

21. Evaluate the integral

( cos(x)√1 + cos(2)² d

Jarcsin(y)

Ivan

y dA

D

where D is the region enclosed by y = 2² and y = 3r.

dx dy