Global extrema of x³ − 3x on [−2, 3]

Critical points. $f'(x)=3x^2-3=0$ at $x=\pm1$, both in $[-2,3]$.

Candidates.
- $f(-2)=-8+6=-2$,
- $f(-1)=-1+3=2$,
- $f(1)=1-3=-2$,
- $f(3)=27-9=18$.

Answer.
- Maximum: $18$, at $x=3$ (an endpoint, not a critical point).
- Minimum: $-2$, attained twice: at $x=-2$ (endpoint) and at $x=1$ (critical point).

Remark. On $\mathbb{R}$ this function has no global extremum (it goes to $\pm\infty$), yet on a segment the extreme value theorem guarantees both. The local maximum $f(-1)=2$ is not the global one.