The two <derivatives> are $3x^2+3$ and $6x$, both equal to six at $x=1$. Their common <tangent line> through $(1,4)$ is $y-4=6(x-1)$, or $y=6x-2$. Setting $x=0$ gives its intersection with the vertical axis:
$$
\boxed{(0,-2).}
$$

Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 10 / b / Solution

 Solution (source code)