8.
$$f ( x ) = ( 8 - 3 x ) ^ { \frac { 4 } { 3 } } \quad 0 < x < \frac { 8 } { 3 }$$
- Show that the binomial expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\) is
$$A - 8 x + \frac { x ^ { 2 } } { 2 } + B x ^ { 3 } + \ldots$$
where \(A\) and \(B\) are constants to be found.
- Use proof by contradiction to prove that the curve with equation
$$y = 8 + 8 x - \frac { 15 } { 2 } x ^ { 2 }$$
does not intersect the curve with equation
$$y = A - 8 x + \frac { x ^ { 2 } } { 2 } + B x ^ { 3 } \quad 0 < x < \frac { 8 } { 3 }$$
where \(A\) and \(B\) are the constants found in part (a).
(Solutions relying on calculator technology are not acceptable.)