6 The diagram below shows a rectangular sheet of metal 24 cm by 9 cm .
\includegraphics[max width=\textwidth, alt={}, center]{10bca9b4-5327-4b35-8b75-612b396e8a76-4_512_897_386_561} A square of side \(x \mathrm {~cm}\) is cut from each corner and the metal is then folded along the broken lines to make an open box with a rectangular base and height \(x \mathrm {~cm}\).

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of liquid the box can hold is given by $$V = 4 x ^ { 3 } - 66 x ^ { 2 } + 216 x$$
2. 1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
   2. Show that any stationary values of \(V\) must occur when \(x ^ { 2 } - 11 x + 18 = 0\).
   3. Solve the equation \(x ^ { 2 } - 11 x + 18 = 0\).
   4. Explain why there is only one value of \(x\) for which \(V\) is stationary.
3. 1. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\).
   2. Hence determine whether the stationary value is a maximum or minimum.