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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]
solving.x1+x3=(a+b)/2....eq1x2+x4=(a-b)/2.....eq2now the meintioned eq can be written as(x1+x3)^2 + (x2+x4)^2 - 2[x1x3 + x2x4]putiing the value of x1 + x3 and x2+x4we 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.x4so as the mentioned eq is minimum,we geta^2+b^2/4 ANS
Plz Approve
(x1^2)+(x2^2)+(x3^2)+(x4^2)
= (x1-x3)^2 + 2x1x3 + (x2-x4)^2 +2x2x4So the expression will have min value when x1=x3 and x2=x4Adding the given eqns, x1+x3= (a+b)/2For min value x1=x3, so x1=x3= (a+b)/4
Similarly, by subtracting the eqns, x2=x4=(a-b)/4Now put the values of x1,x2,x3,x4 in the intial eqn and get the value.
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