> [!note] Theorem > > Suppose that $c$ is a [[Critical Number|critical number]] of a [[Continuity|continuous]] [[Science/Math/Functions/Function|function]] $f$: > - If $f^\prime$ changes from positive to negative at $c$, then $f$ has a [[Extrema#Local|local maximum]] at $c$. > - If $f^\prime$ changes from negative to positive at $c$, then $f$ has a local minimum at $c$. > - If $f^\prime$ has the same sign near $c$, then $f$ has no local extrema at $c$.