# Find the maximum value of $M=\frac{80a^3}{27}+\frac{9b^3}{4}+\frac{abc}{2}$

Let $a,b,c\in \mathbb{R}$ such that: $$3 \ge a \ge b \ge c >0$$

$$2a+3c \ge abc$$

$$\frac{18c}{a}+\frac{4a}{b}+\frac{3b}{c} \ge 3abc$$

Find the maximum value of $M=\frac{80a^3}{27}+\frac{9b^3}{4}+\frac{abc}{2}$

I have tried it for 3 day but I don't get closer.

May be it can be used function or Karamata. I don't know

Could you please give me some suggestions?. Thanks.

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