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Let  \(a,b,c,d\)   be nonnegative real numbers such that \(a + b + c + d = 1\) Find the maximum value of

 

\(a^2 + b^2 + c^2 + d^2\)
 

 Jul 31, 2019
 #1
avatar+6045 
+3

The maximum, 1, will occur when one of the variable is 1 and the rest 0

 

\((a+b+c+d-1)=0\\ (a+b+c+d-1)^2=0\\ a^2+b^2+c^2+d^2 - 2(a+b+c+d)+2(ab+ac+ad+bc+bd+cd)+1 = 0\\ a^2+b^2+c^2+d^2+2(ab+ac+ad+bc+bd+cd)= 1\\ \text{It should be clear that to maximize the sum of the squares we want the second set of terms to be 0}\\ \text{This occurs, as noted above when only 1 element is non-zero}\)

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 Jul 31, 2019
 #2
avatar+133 
+2

Sorry for thanking you late 

 Aug 3, 2019

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