The diagram displays sections of the curves $y = 9 - x^3$ and $y = \frac{8}{x^3}$ together with their intersection points $P$ and $Q$. The $x$-coordinates of $P$ and $Q$ are $a$ and $b$ respectively.
(i)[4]
Show that $x = a$ and $x = b$ are roots of the equation $x^6 - 9x^3 + 8 = 0$. Solve this equation and hence state the value of $a$ and the value of $b$.
(ii)[5]
Find the area of the shaded region bounded by the two curves.
(iii)[4]
The tangents to the two curves at $x = c$ (where $a < c < b$) are parallel. Find the value of $c$.
Worked solution & mark scheme
This 13-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Multiply through by $x^3$ to obtain an equation” …