Application of AM-GM Inequality

Question

If $a,b,c >0$ so that $a+b+c=27$ then what is the maximum value of $(a^2)(b^3)(c^4)$?

I tried the AM-GM inequality but the product term has different powers of $a,b,c$. So how to go about it?

Answer 1

This is a nice Problem! Show that $$a^2b^3c^4\le 544195584$$ and the equal sign holds if $$a=6,b=9,c=12$$

Answer 2

Hint: $$a+b+c=27 \iff \frac{a}{2}+\frac{a}{2}+\frac{b}{3}+\frac{b}{3}+\frac{b}{3}+\frac{c}{4}+\frac{c}{4}+\frac{c}{4}+\frac{c}{4}=27.$$