Question 132
Let $a$, $b$, $c$ be real numbers such that the polynomial $P(x) = x^3 + a x^2 + b x + c$ has three real roots (not necessarily distinct).
Prove that:
$$
12ab + 27c \leq 6a^3 + 10(a^2 - 2b)^{3/2}.
$$
When does equality occur?
Prove that:
$$
12ab + 27c \leq 6a^3 + 10(a^2 - 2b)^{3/2}.
$$
When does equality occur?