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Question

Let a1,a2,a3,a4 be real numbers such that a1+a2+a3+a4=0 and a21+a22+a23+a24=1. Then the smallest possible value of the expression (a1a2)2+(a2a3)2+(a3a4)2+(a4a1)2 lies in the interval
  1. (0,1.5)
  2. (1.5,2.5)
  3. (2.5,3)
  4. (3,3.5)


Solution

The correct option is B (1.5,2.5)
Given :
a1+a2+a3+a4=0 and a21+a22+a23+a24=1
Minimizing the given equation 
(a1a2)2+(a2a3)2+(a3a4)2+(a4a1)2=2[a21+a22+a23+a24]    2[a1a2+a2a3+a3a4+a4a1]=22[a1a2+a3(a2+a4)+a4a1]=22[a1(a2+a4)+a3(a2+a4)]=22[(a2+a4)(a1+a3)]=2+2[(a1+a3)2]
For minimum value of the given expression, (a1+a3)=0
the minimum value of the expression      =2

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