1.21: Past-paper question 9

Figure 4 shows a sketch of part of the curve $\displaystyle C$ with equation $\displaystyle y = f(x)$ where

The curve $\displaystyle C$ has a maximum at the point $\displaystyle A$ with coordinates $\displaystyle \left(-\frac{1}{3}, \frac{100}{27}\right)$ and a minimum at the point $\displaystyle B$ with coordinates $\displaystyle (2, -9)$

Given that $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are integers

(a) show that $\displaystyle a = -5$, $\displaystyle b = -4$ and $\displaystyle c = 3$

(b) (i) Show that $\displaystyle (x+1)$ is a factor of $\displaystyle f(x)$

(ii) Hence, or otherwise, use algebra to factorise $\displaystyle f(x)$ completely.

The curve $\displaystyle C$ crosses the $\displaystyle x$-axis at the points $\displaystyle M$, $\displaystyle N$ and $\displaystyle P$

The finite regions shown shaded in Figure 4 are bounded by the curve $\displaystyle C$ and parts of the $\displaystyle x$-axis from $\displaystyle M$ to $\displaystyle N$ and from $\displaystyle N$ to $\displaystyle P$

(c) Use algebraic integration to determine the total area of the shaded regions.

Give your answer as an exact fraction.