Question 181
For positive real numbers $a$, $b$, $c$ that satisfy $a + b + c = 3$, prove the following:
a.
$$
a\sqrt{b} + b\sqrt{c} + c\sqrt{a} \le 3
$$
b.
$$
\sqrt{a^3 b} + \sqrt{b^3 c} + \sqrt{c^3 a} + \sqrt{ab^3} + \sqrt{bc^3} + \sqrt{ca^3} \le 6
$$
a.
$$
a\sqrt{b} + b\sqrt{c} + c\sqrt{a} \le 3
$$
b.
$$
\sqrt{a^3 b} + \sqrt{b^3 c} + \sqrt{c^3 a} + \sqrt{ab^3} + \sqrt{bc^3} + \sqrt{ca^3} \le 6
$$