6.
$$f ( x ) = - 3 x ^ { 3 } + 8 x ^ { 2 } - 9 x + 10 , \quad x \in \mathbb { R }$$
- Calculate f(2)
- Write \(\mathrm { f } ( x )\) as a product of two algebraic factors.
Using the answer to (a)(ii),
- prove that there are exactly two real solutions to the equation
$$- 3 y ^ { 6 } + 8 y ^ { 4 } - 9 y ^ { 2 } + 10 = 0$$
- deduce the number of real solutions, for \(7 \pi \leqslant \theta < 10 \pi\), to the equation
$$3 \tan ^ { 3 } \theta - 8 \tan ^ { 2 } \theta + 9 \tan \theta - 10 = 0$$