SUNY Geneseo Department of Mathematics

A Lagrange Multiplier Example

Friday, October 28

Math 223 01
Fall 2022
Prof. Doug Baldwin

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Misc

Math Advising

The math department requires second-year majors (so for this year, students who started fall 2021 or spring 2022) to attend an advising meeting. The meetings are advertised as “interactive conversations” aimed at answering students’ questions about planning and completing any of the math majors.

The next (and last) one for this semester is next Wednesday, November 2, at 3:30pm in South 328.

Spring Course Announcement

While it’s not related to calculus, I’ve realized that some of you might have taken an introductory computer programming course (e.g., Math 230, Phys 261, Danl 100) and want to learn more about it.

If so, consider Math 240 this coming spring. It’s a second course in programming, focusing on a kind of programming called “object oriented programming,” and using Python.

It meets Mondays and Wednesdays, 3:30 - 4:45.

I’ll be teaching it.

Lagrange Multiplier Example

What are the maximum and minimum values of f(x,y,z) = x2 - 2y2 + z2 subject to the constraint that x2 + y2 + z2 = 1? You can interpret this geometrically as asking where the maximum and minimum of f(x,y,z) occur given that x, y, and z always have to be coordinates of points on a ball of radius 1 around the origin.

Function definition and constraint equation imply finding minimum and maximum for points lying on sphere

To use Lagrange multipliers, you need the gradients of the objective and constraint functions, which you then use to set up equations from the requirement that f = λg and the constraint:

Gradient equation yields 2 X equals lambda 2 X, negative 4 Y equals lambda 2 Y, 2 Z equals lambda 2 Z

At this point, it wasn’t clear what to do with the equations we’d found. The first 3 imply values for λ, although they disturbingly imply different values (either λ = 1 or λ = -2). The solution to this problem is that each of those equations also has another solution, namely that x, y, or z is 0. So there are actually 2 families of solutions to the equations, one with λ = 1 and the other with λ = -2.

2 X equals lambda times 2 X means lambda is 1 or X is 0; similarly with other equations

Even those 2 families of solutions aren’t enough to get unique values for x, y, or z. Instead, we get implicit equations, whose solutions turn out to be the places where f reaches its minimum and maximum values. In particular, f is maximized at all the points on the circle x2 + z2 = 1, with y = 0, where f(x,y,z) = 1. And f is minimized at the two points where x = z = 0 and y2 = 1, i.e., (0, ±1, 0), where f(x,y,z) = -2.

Lambda equals 1 yields maxima on circle in X Z plane; lambda equals negative 2 yields minima at 1 and negative 1

Next

If multivariable functions have derivatives, you’d think they would have integrals too.

And indeed they do, and they’re the next major subject we’ll talk about.

Geometrically, a 2-variable integral scales the idea of area under a curve to volume under a surface. The idea scales to more dimensions similarly, i.e., to an n-dimensional “volume” under an n-1-dimensional “surface”:

Surface above X Y plane, with tall narrow box extending from plane to surface

To build on this idea, please read “Volumes and Double Integrals,” “Properties of Double Integrals,” and “Iterated Integrals” in section 4.1 of the textbook.

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