# determining the limiting distribution when it's not uniform

$X_1$, $X_2$,... are iid random variables having pdf $$f(x)=3x^2 I_{(0,1)}(x)$$

We also have that $V_n = n^{1/3}$ min$(X_1,...,X_n)$ and $W_n = n^{1/3}$ max$(X_1,...,X_n)$.

a.) Consider the sequence $V_1, V_2,...$ and give the pmf or pdf of the limiting distribution

b.) Consider the sequence $W_1, W_2,...$ and give the pmf or pdf of the limiting distribution

I know how to solve a problem like this when $X_1, X_2,...$ are uniform iid random variables, but I am having trouble figuring out how to start this problem given that the pdf is different.

For part a.) this is what I have so far: The support of $V_n$ is $(0,n^{1/3})$, $$F_{v_n}(v) = P(V_n) \leq v$$ $$=P(n^{1/3} min \{X_1,...,X_n\} \leq v/n^{1/3})$$ $$=1-[P(X_1 > v/n^{1/3})]^n$$ $$=1-[1-P(X_1 \leq v/n^{1/3})]^n$$

I am getting stuck after this trying to determine the cdf and also don't know if this is correct.