6.
$$\mathrm { f } ( x ) = x - [ x ] , \quad x \geq 0$$
where \([ x ]\) is the largest integer \(\leq x\).
For example, \(f ( 3.7 ) = 3.7 - 3 = 0.7 ; f ( 3 ) = 3 - 3 = 0\).
- Sketch the graph of \(y = \mathrm { f } ( x )\) for \(0 \leq x < 4\).
- Find the value of \(p\) for which \(\int _ { 2 } ^ { p } \mathrm { f } ( x ) \mathrm { d } x = 0.18\).
Given that
$$\mathrm { g } ( x ) = \frac { 1 } { 1 + k x } , \quad x \geq 0 , \quad k > 0$$
and that \(x _ { 0 } = \frac { 1 } { 2 }\) is a root of the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\),
- find the value of \(k\).
- Add a sketch of the graph of \(y = \mathrm { g } ( x )\) to your answer to part (a).
The root of \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) in the interval \(n < x < n + 1\) is \(x _ { n }\), where \(n\) is an integer.
- Prove that
$$2 x _ { n } ^ { 2 } - ( 2 n - 1 ) x _ { n } - ( n + 1 ) = 0$$
- Find the smallest value of \(n\) for which \(x _ { n } - n < 0.05\).