Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 6 / b / 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 areThe 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.