Determine whether the function $f(x)=x^3-3x$ is one‑to‑one on the interval $[0,2]$. Justify your answer.

Step-by-step solution

1. Compute the derivative $f^{\prime }(x)=3x^2-3=3(x^2-1)$.
2. On $[0,1)$, $f^{\prime }(x)<0$ so $f$ is strictly decreasing; on $(1,2]$, $f^{\prime }(x)>0$ so $f$ is strictly increasing.
3. The function has a single minimum at $x=1$ but does not attain the same value twice on $[0,2]$.
4. Therefore $f$ is injective (one‑to‑one) on the interval $[0,2]$.

Concept tested: Injectivity on a closed interval

Published 18 September 2026 · How this solution was prepared and checked