I is true if and only if $ -1 < x < 1$; II is true if $x> 1$ or if $x< -1$.

By considering the graph of $y = x - x^2$, which intersects the x - axis at (0,0) and (1,0) and has a maximum at ($\frac{1}{2}, \frac{1}{4}$), it may be seen that statement V is true is and only if $0 < x < 1$.

We see from the table below that a maximum of three statements may be true at any one time.

$x < -1$ then II true- 1

$x=-1$ then none true- 0

$-1< x< 0$ then I, III true- 2

$x=0$ then none true- 0

$0< x< 1$ then I, IV, V true- 3

$x=1$ then none true- 0

$x> 1$ then II true- 1

*This problem is taken from the UKMT Mathematical Challenges.*