<s>[INST] Find the minimum value of $3x + 4y$ The minimum value of  $3x + 4y$  subject to the condition 


$$x^2 y^3  = 6$$ 


and $x$ and $y$ are positive .
 [/INST] Write
$$3x = \frac{3x}{2} + \frac{3x}{2}$$
$$4y = \frac{4y}{3} + \frac{4y}{3}  + \frac{4y}{3} $$
and use $\text{AM} \ge \text{GM}$.
</s>