Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 32A / i

For $x\in [0,1]$, the nonnegative function $x(1-x^2)$ has derivative $1-3x^2$, so its maximum occurs at $x=1/\sqrt{3}$ and equalsThe map preserves the interval exactly when this maximum is at most one:Thus the upper limit is