Define a sequence of functions \(\left\{f_{n}(x)ight\}_{n=1}^{\infty}\) as \(f_{n}(x)=n^{2} x(1-x)^{n}\) for \(x \in \mathbb{R}\) and for all \(n \in \mathbb{N}\).

a. Calculate

\[f(x)=\lim _{n ightarrow \infty} f_{n}(x)\]

for each fixed \(x \in \mathbb{R}\).

b. Calculate

\[\lim _{n ightarrow \infty} \int_{-\infty}^{\infty} f_{n}(x) d x\]

and

\[\int_{-\infty}^{\infty} f(x) d x\]

Is the exchange of the limit and the integral justified in this case? Why or why not?