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\includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-5_710_931_255_607}
The diagram shows parts of the curves \(y = 9 - x ^ { 3 }\) and \(y = \frac { 8 } { x ^ { 3 } }\) and their points of intersection \(P\) and \(Q\). The \(x\)-coordinates of \(P\) and \(Q\) are \(a\) and \(b\) respectively.
- Show that \(x = a\) and \(x = b\) are roots of the equation \(x ^ { 6 } - 9 x ^ { 3 } + 8 = 0\). Solve this equation and hence state the value of \(a\) and the value of \(b\).
- Find the area of the shaded region between the two curves.
- The tangents to the two curves at \(x = c\) (where \(a < c < b\) ) are parallel to each other. Find the value of \(c\).