1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Minimize function subject to constraint

  1. Feb 5, 2014 #1
    Suppose given a function of the form:

    f(x,y,z) = ax + by + cz
    with the constrain x+y=k

    My book minimizes this function by a way I am not completely familiar with:

    dF = adx + bdy + cdz 0

    and since dy=-dx we can write:

    dF = (a-b)dx + cdz = 0
    a-b = c dz/dx (1)

    How I would minimize is simply plug y=k-x into the definition of f:

    f(x,z) = (a-b)x + bk + cz

    And take partial derivatives

    df/dx = a-b + cdz/dx

    df/dz = (a-b) dx/dz + c

    And seting both equal to zero yields a system of equations which does not reduce to (1).
    What is wrong?
  2. jcsd
  3. Feb 5, 2014 #2


    User Avatar
    Homework Helper

    (1) should be
    a-b+c dz/dx=0
    all the equations are equivalent
  4. Feb 5, 2014 #3
    What about the dx/dz term i get?
  5. Feb 5, 2014 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    I hope you have left out some important information, because if we take your problem exactly as you have written it, it has no solution; that is, if a, b, c are all non-zero, you can find a sequence of ##x_n, y_n, z_n## giving ##x_n + y_n = k## for all ##n##, but ##ax_n + by_n+ cz_n \to -\infty ## as ##n \to \infty##. In other words, there is no finite minimum.

    Are you sure you have stated the problem completely and exactly?
    Last edited: Feb 5, 2014
  6. Feb 5, 2014 #5


    User Avatar
    Science Advisor
    Homework Helper

    Which has level sets at z = Ax + B for some suitable constants A and B. I.e. the function is linear and does not have a minimum.

    I would generally tackle these problems using Lagrange multipliers by the way, if you haven't seen them check them out, they are cool ;)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted