let a,b,c real no's such that a+2b+c=4 then find max value of (ab + bc + ac)

(ans=4)

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- Jun 26th 2009, 09:12 AMabhishek arorabasic
let a,b,c real no's such that a+2b+c=4 then find max value of (ab + bc + ac)

(ans=4) - Jun 26th 2009, 10:12 AMTheAbstractionist$\displaystyle a+2b+c=4\,\implies\,b=\frac{4-(a+c)}2$

If you substitute this into $\displaystyle ab+bc+ca$ and simplify, I think you should get

$\displaystyle ab+bc+ca\ =\ 4-\frac{(a-2)^2}2-\frac{(c-2)^2}2$

Hence $\displaystyle ab+bc+ca\,\le\,4$ for all $\displaystyle a,b,c\in\mathbb R.$ - Jun 26th 2009, 10:25 AMabhishek arorafavour
sir can you please add the simplification part actually i could not manage to get to the simplified form

- Jun 26th 2009, 11:56 AMTheAbstractionist
$\displaystyle ab+bc+ca\,=\,b(a+c)+ac$

$\displaystyle =\,\frac{(4-(a+c))(a+c)}2+ac$

$\displaystyle =\,\cdots$

Carry on from there.