$a$, $b$ and $c$ are three sides of a triangle such that $a + b + c = 2$. Calculate the minimum value of $$\large P = 4(a^3 + b^3 + c^3) + 15abc$$
Every task asking for finding the minimum value of an expression containing the product of all of the variables scares me.
Here what I've done.
Using the AM-GM inequality and the Schur's inequality, we have that
$$a^3 + b^3 + c^3 \ge 3abc \implies P \ge \dfrac{9}{2}(a^3 + b^3 + c^3 + 3abc)$$
$$\ge \dfrac{9}{2}[ab(a + b) + bc(b + c) + ca(c + a)] = \dfrac{9}{2}[ab(2 – c) + bc(2 – a) + ca(2 – b)]$$
$$\ge \dfrac{9}{2}[2(ab + bc + ca) – 3abc] \ge \dfrac{27}{2}[2\sqrt[\frac{3}{2}]{abc} – abc]$$
Let $abc = m \implies m \le \left(\dfrac{a + b + c}{3}\right)^3 = \dfrac{8}{27}$
The problem becomes
Find the minimum value of $P' = 2\sqrt[\frac{3}{2}]{m} – m$ when $ 0 < m \le \dfrac{8}{27}$.
which is invalid because there isn't a minimum with the given condition.
Best Answer
Let $a=b=c=\frac{2}{3}$. Thus, $P=8.$
We'll prove that it's a minimal value of $P$.
Indeed, we need to prove that $$\sum_{cyc}(4a^3+5abc)\geq(a+b+c)^3$$ or $$3\sum_{cyc}(a^3-a^2b-a^2c+abc)\geq0,$$ which is true by Schur.