#math/calculus A function $f$ is **increasing** on an interval $I$ if $x_{1}<x_{2}$ implies $f(x_{1})<f(x_{2})$; it is **decreasing** if $x_{1}<x_{2}$ implies $f(x_{1})>f(x_{2})$. **Derivative test for monotonicity.** On any interval where $f$ is differentiable, $f'(x)>0$ makes $f$ increasing and $f'(x)<0$ makes $f$ decreasing. A **critical point** is an interior point $x_{0}$ in the domain where $f'(x_{0})=0$ or $f'(x_{0})$ does not exist. The sign change of $f'$ classifies a critical point: | Sign of $f'$ near $x_{0}$ | Conclusion | |---|---| | $+\to-$ | strict local maximum at $x_{0}$ | | $-\to+$ | strict local minimum at $x_{0}$ | | no sign change | neither | For example, if $f(x)=x^{3}-3x$, then $f'(x)=3(x-1)(x+1)$. Its sign is $+,-,+$ on $(-\infty,-1)$, $(-1,1)$, and $(1,\infty)$, so $f$ has a local maximum $f(-1)=2$ and a local minimum $f(1)=-2$. > [!Note] > A critical point is only a candidate for an extremum. The first-derivative sign change supplies the conclusion.