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Finding the maximum of a product

  1. Aug 10, 2017 #1
    1. The problem statement, all variables and given/known data
    If ##A,B,C,D## are positive numbers such that ##A+2B+3C+4D = 8##, then what is the maximum value of ##ABCD##?

    2. Relevant equations


    3. The attempt at a solution
    From AM-GM, I know that the maximum is ##2/3##. However, I want to know exactly what A,B,C, and D must be to take on this maximum. From AM-GM, I know that it must be the case that A=2B=3C=4D for equality to occur, but I'm not sure how I can use this explicitly, as a four-way equality is unwieldy.
     
  2. jcsd
  3. Aug 10, 2017 #2

    fresh_42

    Staff: Mentor

    What is the problem? If you have ##A=2B=3C=4D## then ##8=A+2B+3C+4D=4A=(4\cdot 2)B=(4\cdot 3)C=(4\cdot 4)D##.
     
  4. Aug 11, 2017 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    This is a standard constrained multivariate optimization problem, and can be solved using a Lagrange multiplier method, for example.
     
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