Chapter 8, Problem 1

[dstefan]

Chapter 8, Problem 1:

Find the maximum and minimum of the function f(x_1,x_2,x_3) = 4x_2 - 2x_3 subject to the constraints 2x_1 - x_2 - x_3 = 0 and x_1^2 + x_2^2 = 13.


Solution:

a

[maxsidious]

Thank you soooo much
Where can I get the answer?

[doug reich]

I think the answer is forthcoming... but you should probably use Laplace multipliers. That is let

g(x_1,x_2,x_3,\lambda,\gamma) = 4x_2-2x_3 + \lambda (2x_1-x_2-x_3)+\gamma (x_1^2+x_2^2-13)

Then take the partial derivative of g w.r.t. each of x_1,x_2,x_3,\lambda,\gamma, set each of the derivatives equal to zero, and solve the resulting system to find the x's (in the process you'll probably find \lambda \text{ and } \gamma as well.

The easier way may be to observe what kind of objects (ie are those functions cones, spheres, planes, ellipsoids, cylinders, etc.) you're dealing with, and draw a picture or use intuition to figure out where the extrema are (or a combination of intuition and calculus, anyway).

[Sanket Patel]

maxsidious

If you are unfamiliar with inequality constrained optimization, take a look at these (http://garnet.acns.fsu.edu/%7Etsalmon/ineqnotes.pdf) lecture notes. The notes are written in a very lucid manner and should be very easy to follow. The notes will give a bit of intuition behind the process Doug has outlines. It really is quite straightforward - its more algebra than calculus.