Question

Use the method of Lagrange multipliers to find the minimum value of the function

f(x,y,z)=x2+y2+z2

subject to the constraints x+y=10 and 2y−z=3.

Answer #1

Find the minimum of f(x,y,z) = x2 + y2 +
z2 subject to the two constraints x + 2y + z = 3 and x -
y = 4 by answering following questions
a) write out the lagrange equation involving lagrange
multipliers λ(lamba) and μ(mu)
b) solve for lamba in terms of x and y
c) solve for x,y,z using the constraints
d) determine the minimum value

Use Lagrange multipliers to find the extremal values of
f(x,y,z)=2x+2y+z subject to the
constraint
x2+y2+z2=9.

Use the method of Lagrange Multipliers to find the maximum
value:
f(x,y,z) = x2y2z2 subject to
the constraint x2+y2+z2=1 no
decimals permitted

Use Lagrange Multipliers to find the extreme values of f(x,y,z)
= x2 + 3y subject to the constraints x2 +
z2 = 9 and 3y2 + 4z2 = 48.

Find the minimum of f(x, y, z) = x2 + y2 +
z2 subject to the two constraints x + y + z = 1 and 4x +
5y + 6z = 10

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 the method of Lagrange multipliers to find the maximum and
minimum values of F(x,y,z) = 5x+3y+4z, subject to the constraint
G(x,y,z) = x2+y2+z2 = 25. Note the
constraint is a sphere of radius 5, while the level surfaces for F
are planes. Sketch a graph showing the solution to this problem
occurs where two of these planes are tangent to the sphere.

Use Lagrange multipliers to find the maximum and minimum values
of f(x,y)=x2+5y
subject to the constraint x2-y2=3 , if
such values exist.
Maximum =
Minimum

Using Lagrange multipliers, find the coordinates of the minimum
point on the graph of z=x2+y2 subject to the constraint
2x+y=20.
Lagrange function (use k for lambda) L(x,y,k)=
Lx(x,y,k)=
Ly(x,y,k)=
Lk(x,y,k)=
Minimum Point (format (x,y,z)):

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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