Let `a ,ba n dc` be real numbers such that `a+2b+c=4` . Find the maximum value of `(a b+b c+c a)dot`

Explanation

Given `a+2b+c=4`
`impliesa=4-2b-c`
Let `y=ab+bc+ca=a(b+c)+bc`
`=(4-2b-c)(b+c)+bc`
`=-2b^(2)+4b-2bc+4c-c^(2)`
`implies2b^(2)+2(c-2)b-4c+c^(2)+y=0`
Since `b epsilonR` so
`4(c-2)^(2)-4xx2xx(-4c+c^(2)+y)ge0`
`implies(c-2)^(2)+8c-2c^(2)-2yge0`
`impliesc^(2)-4c+2y-4le0`
Since `c epsilonR` so `16-4(2y-4)ge0impliesyle4`
Hence maximum value of `ab+bc+ca` is 4.
Aliter
`:'AMgeGM`
`implies((a+b)+(b+c))/2gesqrt((a+b)(b+c))`
`implies2gesqrt((ab+bc+ca+b^(2)))[:'a+2b+c=4]`
`impliesab+bc+cale4-b^(2)`
`:.` Maximum value fo `(ab+bc+ca)` is 4.