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 or a = 4 - 2b -c`
Let `ab + bc + ca = x or a(b +c) + be = x`
or `(4 - 2b - c) (b + c) + bc = x`
or `4b + 4c - 2b^(2) - 2bc - bc - c^(2) + be = x `
or `2b^(2) - 4b + 2bc - 4c + c^(2) + x = 0`
or ` 2b^(2) + 2(c - 2) b - 4c + c^(2) + x = 0`
Since b `in` R, so
` 4(c - 2)^(2) - 4xx2(-4c + c^(2) + x) ge 0`
or `c^(2) - 4c + 4 + 8c - 2c^(2) - 2x ge 0`
or `c^(2) - 4c + 2x - 4 le 0`
Since c `in` R, so
`16 - 4 (2x - 4) ge 0 rArr x le 4 `
`therefore` max ( ab + bc + ac )= 4`