= Solution
The full <cumulative distribution function> is
$$
\boxed{F(x)=\begin{cases}0,&x<0,\\4x^3-3x^4,&0\leq x\leq1,\\1,&x>1.\end{cases}}
$$
Thus the <function graph> is horizontal at zero for $-1\leq x\leq0$, rises continuously from $(0,0)$ to $(1,1)$, and is horizontal at one for $1\leq x\leq2$. Its <derivative> $12x^2(1-x)$ is positive on $(0,1)$ and zero at both endpoints. The second <derivative> $12x(2-3x)$ changes sign at $x=2/3$, giving an inflection <point> there.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1-q16-distribution.png]
{title=Cumulative distribution and density, with the mode marked, for Question 16}
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