<s>[INST] Maximum value of equation $ab+bc+ca$ if $a+2b+c=4$ If $a,b,c$ ${\in}$ $\mathbb{R} $ such that
$a+2b+c=4$, then find the $max$ value of
$ab+bc+ca$.
I always get stuck with max, min questions. We cannot apply AM:GM here, I have not studied much  calculus yet.
Can you do this with graphs or by plane algebra?
I don't really know what tag to put for this!
 [/INST] $$a+b+c=4-b$$
$$a^2+b^2+c^2+2(ab+ac+bc)=16-8b+b^2$$
$$ab+bc+ca=\frac{16-8b-a^2-c^2}2$$
Since $b=\frac{4-a-c}2$,
$$ab+bc+ca=\frac{16-16+4a+4c-a^2-c^2}2=\frac{a(4-a)+c(4-c)}2\le\frac{4+4}2=4$$
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