Finding the min value of an expression

Which area in math you prefer to investigate further?

0 vote(s)
0.0%

2 vote(s)
100.0%

0 vote(s)
0.0%

0 vote(s)
0.0%

0 vote(s)
0.0%
6. Probability

0 vote(s)
0.0%
1. Jan 21, 2012

superconduct

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

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

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

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

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. Jan 21, 2012

Joffan

Your first approach is good, but try subtracting instead of adding. Then you can demonstrate (as you have) that the lowest answer is achievable.