confusion regarding minimum value

Question

Let $a+2b+c=4$.

Find the maximum value of $ab+bc+ac$.

I tried using the arithmetic-geometric mean concept, but couldn't get to the answer. The Cauchy-Schwartz equation isn't of great help either.

Answer 1

Eliminating $b,$

$$ab+bc+ca=b(a+c)+ca=\dfrac{4-(c+a)}2\cdot(c+a)+ca$$

$$=\dfrac{4(c+a)-c^2-a^2}2$$

$$=\dfrac{8-(c-2)^2-(a-2)^2}2\le\dfrac82$$

Answer 2

Solve the first equation for $a,$ substitute into the second equation and find the maximum of that function (of two variables $b$ and $c$) by your favourite method. Since it is quadratic you could re-write it in standardized form.