# 1

The function f(x,y) = 4x + 4y has an absolute maximum value and absolute minimum value subject to the

constraint 25x² - 25xy + 25y² = 1. Use Lagrange multipliers to find these values.

The absolute maximum value is (Type an exact answer.)

#2

The function f(x,y) = 5xy has an absolute maximum value and absolute minimum value subject to the constraint

2x² + 2y² - 3xy = 49. Use Lagrange multipliers to find these values.

The absolute maximum value is

#3

5xy

The function f(x,y) = e has an absolute maximum value and absolute minimum value subject to the

constraint x² + xy + y² = 36. Use Lagrange multipliers to find these values.

The absolute maximum is

(Type an exact answer in terms of e.)

#4/n# 5

The function f(x,y,z) = 7x has an absolute maximum value and absolute minimum value subject to the constraint

x² + y² +2²-z=2. Use Lagrange multipliers to find these values.

The maximum value is

# 6

The function f(x,y,z) = 6x + z² has an absolute maximum value and absolute minimum value subject to the

constraint 3x² + 2y² +22² =27. Use Lagrange multipliers to find these values.

The absolute maximum value is

#7

Find the point on the plane 2x + 3y +6z-16=0 closest to the point (-3,4,2).

The point is). (Simplify your answers.)

# 8

Evaluate the iterated integral 24x³y dy d

| | 24x²y³dy dx =

-4 0

C/n# 9

Evaluate the following iterated integral.

20

x siny dy dx

x siny dy dx =

20

(Simplify your answer.)

# 10

Evaluate the following double integral over the region R by converting it to an iterated integral.

SS 3x+4y dA: R={(x,y): 0≤x≤ In 2, 1sys In 4)

Evaluate the integral.

SS² 3x+4y

"dA= (Type an exact answer.)

R

# 11

When converted to an iterated integral, the following double integral is easier to evaluate in one order than the

other. Find the best order and evaluate the integral.

[√xsec³(x)dA; R=(x,y): 0≤x≤ 3.0y≤1)

R

Select the correct answer below and fill in the answer box to complete your choice./n# 12

Compute the average value of the following function over the region R.

f(x,y) = 7 sinxcosy R={(xy): 0sxs₁0sys}

(Simplify your answer. Type an exact answer, using radicals as needed. Type your answer in factored form.

Use integers or fractions for any numbers in the expression.)

# 13

Find the average squared distance between the points of R={(x,y): 0≤x≤4,0 ≤ y ≤ 3) and the point (4,3).

The average squared distance is

(Type an integer or a simplified fraction.)

# 14

Evaluate the following iterated integral.

01

9y

9y

dx dy

C

dx dy=

01

(Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)