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.