Real numbers , , , , and satisfy the system of equations below:
Find the sum of the maximum and minimum values of .
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b + c + d + e = ( 8 − a )
The Artithematic mean is : x = 4 ( 8 − a ) Let b = x + b 1 , c = x + c 1 , d = x + d 1 and e = x + e 1
Then b 1 + c 1 + d 1 + e 1 = 0 and
1 6 = a 2 + 4 x 2 + b 1 2 + c 1 2 + d 1 2 + e 1 2 ≥ a 2 + 4 x 2 = a 2 + 4 ( 8 − a ) 2
or a ( 5 a − 1 6 ) ≤ 0
And Hence the max and min of a are 1 6 / 5 and 0 respectively