Problem

Figure 2 shows a sketch of the curve $C$ with equation $y=x^2-5x+13$.

Figure 2

The point $M$ is the minimum point of $C$.

The straight line $l$ passes through the origin $O$ and intersects $C$ at the points $M$ and $N$ as shown.

Find, showing your working,

(a) the coordinates of $M$,

(b) the coordinates of $N$.

Figure 3 shows the curve $C$ and the line $l$. The finite region $R$, shown shaded in Figure 3, is bounded by $C$, $l$ and the $y$-axis.

Figure 3

(c) Use inequalities to define the region $R$.