# If x1+x2+x3+x4=a and x1-x2+x3-x4=b .  Then find the minimum value of  (x1^2)+(x2^2)+(x3^2)+(x4^2) in terms of a and b.      [ANS (a^2+b^2)/4] . Plz tell the process. [note here x1 doesnot mean x*1. it means 1 is a suffix]

Akash Kumar Dutta
98 Points
10 years ago

solving.
x1+x3=(a+b)/2....eq1
x2+x4=(a-b)/2.....eq2
now the meintioned eq can be written as
(x1+x3)^2 + (x2+x4)^2 - 2[x1x3 + x2x4]
putiing the value of x1 + x3  and x2+x4
we have=a^2+b^2/2 - 2 [x1.x3 + x2.x4].
now apply AM>GM in the first two equations and get the max value of x1.x3 and x2.x4
so as the mentioned eq is minimum,we get
a^2+b^2/4 ANS

Plz Approve

mycroft holmes
272 Points
10 years ago

Saikat Bhaumik
4 Points
10 years ago

 (x1^2)+(x2^2)+(x3^2)+(x4^2)

= (x1-x3)^2 + 2x1x3 + (x2-x4)^2 +2x2x4

So the expression will have min value when x1=x3 and x2=x4

Adding the given eqns,  x1+x3= (a+b)/2

For min value x1=x3, so x1=x3= (a+b)/4

Similarly, by subtracting the eqns, x2=x4=(a-b)/4

Now put the values of x1,x2,x3,x4 in the intial eqn and get the value.