(adsbygoogle = window.adsbygoogle || []).push({}); 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**