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Question

Let a, b, c be real number such that a+2b+c=4. Find max (ab+bc+ca)

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Solution

Consider the problem
Given,
a,b,c are real number
And also, a+2b+c=4
a+c=42b...........(1)
Max(ab+bc+ca)
=max(b(a+c)+ca)=max(b(42b)+ca)=max(4b2b2+ca)
This is maximum when ac>0 and their sum is constant such that
a=c=(42b2)=2b
Max(4b2b2+(2b)2)=max(4b2b2+44b+b2)=max(4b2)=4[b2>0]

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