Algebra equation with 4 variables
Algebra
Level
3
If a+b+c+d=1 then what's the maximum value of (ab+bc+cd) if a,b,c,d are non negative. Kindly round of your answer to 2 decimal places.
Note : A similar question was asked in the 2013 Australian Intermediate Olympiad.
The answer is 0.25.
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1 solution
If a+b+c+d=1 then what's the maximum value of (ab+bc+cd) if a,b,c,d are greater than 0. Kindly round of your answer to 2 decimal places.
(a+c)(b+ d) = ab + ad +bc + cd , therefore ab +bc + cd = ab+ad+bc+cd - ad = (a+c)(b+d) -ad.
If we have find out the maximum value of (a+c)(b+d) - ad , this occurs when (a+c)(b+d) is maximum and ad is 0.
So let x = (a+c) and y = (b+d) , the original problem translates to finding the maximum of value xy given that x+y = 1 (as ad = 0).
This occurs when x = 0.5 and y = 0.5. or when xy = 0.25