1.1: Turning points of a quartic curve

The point with coordinates $\displaystyle (4,-104)$ lies on the curve $\displaystyle C$ with equation $\displaystyle y=f(x)$.

Given that

(a) (i) show that $\displaystyle C$ passes through the origin,

(ii) show that $\displaystyle C$ has a maximum at the point on the curve where $\displaystyle x=0.5$.

The curve $\displaystyle C$ has another turning point at $\displaystyle A$ and another turning point at $\displaystyle B$. Given that the $\displaystyle x$ coordinate of $\displaystyle A$ is negative,

(b) (i) find the coordinates of $\displaystyle A$ and the coordinates of $\displaystyle B$,

(ii) determine the nature of these turning points.