Question

Find the minimum of f(x,y,z) = x^{2} + y^{2} +
z^{2} 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

Answer #1

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.

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

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.

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

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

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 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.

Calculate ∫ ∫S f(x,y,z)dS for the given surface and function.
x2+y2+z2=144, 6≤z≤12; f(x,y,z)=z2(x2+y2+z2)−1.

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