As we know

AM  $$ \ge $$  GM

$$ \Rightarrow $$   $${{3\left( {{x \over 3}} \right) + 4\left( {{y \over 4}} \right) + 5\left( {{z \over 5}} \right)} \over {12}}$$  $$ \ge $$  $${\left[ {{{\left( {{x \over 3}} \right)}^3}{{\left( {{y \over 4}} \right)}^4}{{\left( {{z \over 5}} \right)}^5}} \right]^{{1 \over {12}}}}$$

$$ \Rightarrow $$   1  $$ \ge $$  $${{{x^3}{y^4}{z^5}} \over {{3^3}{4^4}{5^5}}}$$

$$ \Rightarrow $$   x³ y⁴ z⁵  $$ \le $$  3³ . 4⁴ . 5⁵

$$ \Rightarrow $$   x³ y⁴ z⁵  $$ \le $$  (0.1)(600)³

but given that,

x³ y⁴ z⁵ = (0.1) (600)³

$$ \therefore $$   AM  $$=$$  GM

$$ \Rightarrow $$   All the number are equal.

$$ \therefore $$   $${x \over 3} = {y \over 4} = {z \over 5} = k$$

$$ \Rightarrow $$  x $$=$$ 3k, y = 4k, z = 5k

given that,

        x + y + z $$=$$ 12

$$ \Rightarrow $$   3k + 4k + 5k $$=$$ 12

$$ \Rightarrow $$   12k $$=$$ 12

$$ \Rightarrow $$   k = 1

$$ \therefore $$   x $$=$$   3,  y $$=$$ 4,   z $$=$$ 5

So, x³ + y³ + z³

$$=$$ 3³ + 4³ + 5³

$$=$$ 216