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: Finding the min value of an expression

  1. Algebraic Number Theory, Algebraic Geometry, and Representation Theory

    0 vote(s)
  2. Topology and Geometry

    2 vote(s)
  3. Applied Analysis, Control Theory and Game Theory

    0 vote(s)
  4. Combinatorics

    0 vote(s)
  5. Scientific Computing and Numerical Analysis

    0 vote(s)
  6. Probability

    0 vote(s)
  1. Jan 21, 2012 #1
    1. The problem statement, all variables and given/known data
    [itex]a,b,c,d [/itex] are real numbers
    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^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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook