Review question

# When is this integral less than or equal to $1 - A/(n+B)$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8396

## Question

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

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

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

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

4. 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 $A \le \dfrac{3}{4}$.

5. When $A=\dfrac{3}{4}$, what is the smallest possible value of $B$?