Maximum value 2

[rsoldo]

Let $a,b,c$ positive real numbers such that $a+b+c=1$. Determine the maximum value of
$S=\sqrt[3]{a+b}+\sqrt[3]{b+c}+\sqrt[3]{c+a}$.

[maxmatem]

AM-GM ??

[zykov]

$$3\sqrt[3]{\frac23}=\sqrt[3]{18}$$

[TOTAL]

$\sqrt[3]{a+b}+\sqrt[3]{b+c} \le 2\sqrt[3]{b+c/2+a/2}$, т.к. $f^{''}(x) < 0, \; f(x) = \sqrt[3]{x}$