Question

**Calculus III. Please show all work and mark the
answer(s)!**

1) Use Lagrange multipliers to find the maximum and minimum values of the function f(x, y) = x^2 + y^2 subject to the constraint xy = 1.

2) Use Lagrange Multipliers to find the point on the curve 2x + 3y = 6 that is closest to the origin. Hint: let f(x, y) be the distance squared from the origin to the point (x, y), then find the minimum of f subject to the constraint that the point (x, y) must be on the given curve.

**Thank you in advance.**

Answer #1

1. Use the method of Lagrange multipliers to find the
maximize
of the function f (x, y) = 25-x^2-y^2 subject to the constraint
x + y =-1
2. Use the method of Lagrange multipliers to find the
minimum
of the function f (x, y) = y^2+6x subject to the constraint
y-2x= 0

Use Lagrange multipliers to find all relative extrema of the
function subject to the given constraint.
f(x,y)=x^2+2y^3
constraint: 2x+y^2-8=0

Use the Lagrange Multipliers method to find the maximum and
minimum values of f(x,y) = xy + xz subject to the constraint x2 +y2
+ z2 = 4.

Use Lagrange multipliers to find both the maximum and minimum of
the function f(x, y) = 3x + 4y subject to the constraint that the
point be on the circle x 2 + y 2 = 100.

Use the method of Lagrange multipliers to find the extreme
values of f(x,y)=
x^(2) + y^(2) − xy − 4 subject to the constraint x + y = 6.

Use the method of Lagrange Multipliers to find the extreme
value(s) of f(x, y) = 3x + 2y subject to the constraint y = 3x ^2 .
Identify the extremum/extrema as maximum or minimum.

Use the method of Lagrange multipliers to find the maximum value
of f(x,y) = xy subject to the constraint x^2=y^2=7 (you may assume
that the extremum exists)

Use Lagrange Multipliers to find both the maximum and minimum
values of f(x, y) = 4xy subject to the constraint x^2 + y^2 =
2.

Use Lagrange multipliers to find the maximum and minimum values
of
f(x,y)=xy
subject to the constraint 25x^2+y^2=200
if such values exist.
Enter the exact answers. Which is global maximum/global minimum?
Enter NA in the appropriate answer area if these do not apply.

Use Lagrange multipliers to find the minimum value of f ( x , y
, z ) = 2 x 2 + y 2 + 3 z 2 subject to the constraint 2 x − 3 y − 4
z = 49. The minimum value of f ( x , y ) subject to 2 x − 3 y − 4 z
= 49 is

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