You are given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x - 4 )\).
(A) Sketch the graph of \(y = \mathrm { f } ( x )\).
(B) Show that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
You are given that \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 40\).
(A) Show that \(\mathrm { g } ( 5 ) = 0\).
(B) Express \(\mathrm { g } ( x )\) as the product of a linear and quadratic factor.
(C) Hence show that the equation \(\mathrm { g } ( x ) = 0\) has only one real root.
Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).