How find this maximum $Aa^2+Bb^2+Cc^2$ if $a+b+c=1$ let $a+b+c=1,a,b,c\ge 0$,find this following maximum
$$f(a,b,c)=Aa^2+Bb^2+Cc^2$$
where $A,B,C$ be postive constant numbers.
My idea: if find this minimum value,I can find it,because we have use Cauchy-Schwarz inequality,then we have
$$(Aa^2+Bb^2+Cc^2)(\dfrac{1}{A}+\dfrac{1}{B}+\dfrac{1}{C})\ge (a+b+c)^2=1$$
so
$$Aa^2+Bb^2+Cc^2\ge\dfrac{ABC}{AB+BC+AC}$$
But find this  maximum,I can't to solve it ,Thank you for you help,

Let $M=\text{max}\{A.B.C\}$, then
$$
Aa^2+Bb^2+Cc^2\le M(a^2+b^2+c^2)\le M(a+b+c)^2=M
$$
Obviously $M$ is attained, because $A,B$ and $C$ are attained, hence $M$ is the maximum value of your function.
