MATH224: Studio 7 Assignment (Lagrange Multipliers)
Created by: Gus Greivel
Last Modified: 10/8/2015
Background:
We discussed the method of Lagrange Multipliers (Wikepedia – Lagrange Multiplier) in class
this week. This method allows us to find points that may correspond to the absolute maximum
or minimum values of an objective function f(x,y,z) subject to one or more constraints stated in
the form of g(x, y, z) = k .
Specifically, if f and g are differentiable functions and
on the surface g(x, y, z) = k , then:
I.
Find all values x, y, z, and  such that Ñf (x, y, z) = lÑg(x, y, z) and g(x, y, z) = k .
II.
Evaluate f(x,y,z) at all points (x,y,z) that result from step I. The largest of these values
is the absolute maximum value of f on the constraint; the smallest is the absolute
minimum value of f on the constraint.
Practice Problems:
Use the method of Lagrange Multipliers to find the absolute extremes requested in each of the following
problems. Include the following for each:
(a)
(b)
(c)
(d)
(e)
Identify the objective function and the constraint. Write the constraint in the form g(x, y, z) = k .
Find the gradients and set up the complete system of equations to be solved.
Solve this system and list all points that satisfy it.
Evaluate the objective function at all points identified in (c) and determine the maximum and
minimum values of the function.
Modify and use the Mathematica notebook for this studio to validate your work.
1. Find the maximum and minimum values of the hyperbolic paraboloid, z = y2 – x2 on the unit circle
x2 + y2 = 1.
2. A storage shed is to be constructed in the shape of a rectangular prism. Assume the floor is three
times as expensive as the walls and that the roof is twice as expensive at the walls. Assuming that
the shed must have a capacity of 2500 ft3, determine the dimensions that will minimize the cost of
this shed.
3. You are in charge of erecting a radio telescope on a newly discovered planet. To minimize
interference, you want to place it where the magnetic field on the planet is weakest. The planet is
spherical, with a radius of 6 units. Based on a coordinate system whose origin is at the center of the
planet, the strength of the magnetic field is given by M(x, y, z) = 6x - y 2 + xz +80 . Where should you
locate the telescope?
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Two Constraint Problem:
If we have two constraints g(x, y, z) = k and h(x, y, z) = c that are linearly independent (i.e.
their gradients are not parallel) with
and
, the extreme values of f (x, y, z)
subject to these two constraints will necessarily occur where Ñf is parallel to both Ñg and Ñh .
Specifically, Ñf is in the plane determined by Ñg and Ñh at any such point. This means that
there will be numbers (Lagrange Multipliers)  and  such that Ñf = lÑg + mÑh at any points
corresponding to extremes.
This leads to the following modification of the process described above:
I.
Find all values x, y, z,  and  such that Ñf (x, y, z) = lÑg(x, y, z)+ mÑh(x, y, z) and
g(x, y, z) = k and h(x, y, z) = c .
II.
Evaluate f(x,y,z) at all points (x,y,z) that result from step I. The largest of these values
is the absolute maximum value of f on the constraint; the smallest is the absolute
minimum value of f on the constraint.
Practice Problem:
Use the method of Lagrange Multipliers with two constraints to find the absolute extremes requested in
the following problem. Include the following:
(a)
(b)
(c)
(d)
(e)
Identify the objective function and the constraints. Write the constraints in the form
g(x, y, z) = k and h(x, y, z) = c .
Find the gradients and set up the complete system of equations to be solved.
Solve this system (you may want to use the Mathematica notebook for this) and list all points
that satisfy it.
Evaluate the objective function at all points identified in (c) and determine the maximum and
minimum values of the function.
Comment on the graphs included for this problem in the Mathematica notebook.
4. Consider the curve of intersection of the paraboloid z = x 2 + 2y 2 and the cylinder x 2 + y 2 =1 . Find
the point on this curve that is closest to the point (2, -2,3).
Reflection Question:
Suppose you had an objective function of four variables f (w, x, y, z) = wx + x 2 + 3y2 + z3 with
three constraints: g1 (w, x, y, z) = x 2 + y2 + z 2 = 4, g2 (w, x, y, z) = wxyz = 3 and
g3 (w, x, y, z) = wx + xy + yz = 2 . How would you generalize the method of Lagrange Multipliers to
solve this problem? Write out the system of equations to be solved to identify critical points for
this problem. (DON’T TRY TO FIND THE CRITICAL POINTS.)
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