= Solution
The <function graph> is a horizontally shifted and scaled fourth power. It is nonnegative and touches the horizontal axis at its fourfold <polynomial root>. The axis intersections are
$$
\boxed{(-1/2,0)\quad\text{and}\quad(0,1).}
$$
The <derivative> is $8(1+2x)^3$, so the graph decreases to its flat <local minimum> at $x=-1/2$ and then increases. Its endpoint values on the requested interval are $f(-1)=1$ and $f(1)=81$. The original sketch includes a closer view of the minimum so that the small values are visible beside the much larger right endpoint.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-1-quartic.png]
{title=The fourth-power polynomial on the requested interval, with a close view of its flat minimum}
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