For a positive whole number \(n\), the function \(f_{n}(x)\) is defined by \[f_{n}(x) = (x^{2n-1}-1)^2.\]

Sketch the graph of \(y=f_{2}(x)\) labelling where the graph meets the axes.

On the same axes sketch the graph of \(y=f_{n}(x)\) where \(n\) is a large positive integer.

Determine \[\int_0^1 \! f_{n}(x) \, \mathrm{d}x.\]

The

*positive*constants \(A\) and \(B\) are such that \[\int_0^1 \! f_{n}(x) \, \mathrm{d}x \le 1 - \frac{A}{n+B} \quad \text{for all $n \ge 1$}.\]Show that \[(3n-1)(n+B) \ge A(4n-1)n,\] and explain why \(A \le \dfrac{3}{4}\).

When \(A=\dfrac{3}{4}\), what is the smallest possible value of \(B\)?