Problem
3 The diagram shows the curve $y=\mathrm{f}(x)$, where $\mathrm{f}(x)$ is a cubic polynomial in $x$. This diagram is repeated in the Printed Answer Booklet. (a) State the values of $x$ for which $f(x)<\frac{1}{2}$, giving your answer in set notation. [3] [2] (b) On the diagram in the Printed Answer Booklet, draw the graph of $y=\mathrm{f}(-x)$. (c) Explain how you can tell that $\mathrm{f}(x)$ cannot be expressed as the product of three real linear [1] factors.
Solution
Step 1 ：\(g(x) = f(x) - \frac{1}{2}\)
Step 2 ：\(g(x) = cx^3 + bx^2 + ax - \frac{1}{2}\)
Step 3 ：Find the values of x for which \(g(x) < 0\)
Step 4 ：Express the answer in set notation: \(\boxed{\{x \mid x \in (a, b)\}}\)
