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Finding the min value of an expression

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  1. Jan 21, 2012 #1
    1. The problem statement, all variables and given/known data
    [itex]a,b,c,d [/itex] are real numbers
    [itex]ab+bc+cd+da=16[/itex]
    Find the minimum value possible for [itex]a²+b²+c²+d²[/itex]


    2. Relevant equations
    [itex]ab+bc+cd+da = (a+c)(b+d)[/itex]
    [itex]x^2 \ge 0[/itex] for some real x

    3. The attempt at a solution
    1st approach:[itex]2(a^2 +b^2 +c^2 +d^2) + 2(ab+bc+cd+da)[/itex] [tex]= (a+b)^2 +(b+c)^2 +(c+d)^2 +(d+a)^2 \ge 0[/tex]
    which gives [itex]a^2 +b^2 +c^2 +d^2 \ge -16[/itex] that cannot give the min. value.

    2nd approach: [itex] (a+c)(b+d) =16[/itex]
    By differential calculus, rectangle of fixed area has min. diagonal when it is square,
    [itex]a+c=b+d=4[/itex]
    [itex]a=b=c=d=2[/itex]
    [itex]a^2 +b^2 +c^2 +d^2 \ge 2^2+2^2+2^2+2^2 =16[/itex]

    How would you solve this? If you think there is any unclear reasonings in my post or others' replies please do not mind elaborating/criticizing/replacing them for me and others.
     
  2. jcsd
  3. Jan 21, 2012 #2
    Your first approach is good, but try subtracting instead of adding. Then you can demonstrate (as you have) that the lowest answer is achievable.
     
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