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

\(a^2 + b^2 + c^2 + d^2\)

Havingfun Jul 31, 2019

#1**+3 **

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

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

.Rom Jul 31, 2019