let a,b,c real no's such that a+2b+c=4 then find max value of (ab + bc + ac)
(ans=4)
$\displaystyle a+2b+c=4\,\implies\,b=\frac{4-(a+c)}2$
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 $\displaystyle a,b,c\in\mathbb R.$