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Equations -maximum

  1. Dec 28, 2012 #1
    Hey!
    I found one difficult equation for me...
    It is:
    Valid for real numbers a, b, c, d:
    a + b = c + d
    ad = bc
    ac + bd = 1
    What is the maximum of a+b+c+d?

    And DON'T SAY wolfram, really no wolfram... I used it and it isn't good.
    So... have you got any idea?
     
  2. jcsd
  3. Dec 28, 2012 #2
    Add ##a##c to both sides of the second equation, and you get
    $$
    ad + ac = ac + bc
    \iff
    a(c+d) = (a+b)c = (c+d)c.
    $$
    One possibility now would be ##c+d=0##, but then you'd also have ##a+b=0##, and so ##a+b+c+d=0##. So suppose ##c+d\neq0##, then ##a=c##, and from the first equation you also have ##b=d##.

    Now you're really left with only two unknowns, which you can use in the third equation to make ##a+b+c+d=2a+2b## minimal.
     
  4. Dec 28, 2012 #3
    Hmm... Sorry, bud I don't understand last line. Ok, a+b+c+d=2a+2b, yes, but how I find numerical value of a+b+c+d in second variant?
     
  5. Dec 28, 2012 #4
    Your third equation says ##ac+bd=1##, i.e. ##a^2+b^2=1##, so if you assume ##b\geq0## (since you want ##2a+2b## to be maximal), you get ##b=\sqrt{1-a^2}##. This means you need to find a value ##a## for which ##2a+2b = 2a+2\sqrt{1-a^2}## is maximal.

    Do you know how to find such an ##a##?
     
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