Editing Derivative test (section)

===Precise statement of first-derivative test===
The first-derivative test depends on the "increasing–decreasing test", which is itself ultimately a consequence of the [[mean value theorem]]. It is a direct consequence of the way the [[derivative]] is defined and its connection to decrease and increase of a function locally, combined with the previous section.

Suppose ''f'' is a real-valued function of a real variable defined on some [[interval (mathematics)|interval]] containing the critical point ''a''. Further suppose that ''f'' is continuous at ''a'' and [[differentiable function|differentiable]] on some open interval containing ''a'', except possibly at ''a'' itself.

* If there exists a positive number ''r''&nbsp;>&thinsp;0 such that for every ''x'' in (''a'' − ''r'', ''a'') we have {{nobr|''{{′|f}}''(''x'') ≥ 0,}} and for every ''x'' in (''a'', ''a'' + ''r'') we have {{nobr|''{{′|f}}''(''x'') ≤ 0,}} then ''f'' has a local maximum at ''a''.
* If there exists a positive number ''r''&nbsp;>&thinsp;0 such that for every ''x'' in (''a'' − ''r'', ''a'') we have {{nobr|''{{′|f}}''(''x'') ≤ 0,}} and for every ''x'' in (''a'', ''a'' + ''r'') we have {{nobr|''{{′|f}}''(''x'') ≥ 0,}} then ''f'' has a local minimum at ''a''.
* If there exists a positive number ''r''&nbsp;>&thinsp;0 such that for every ''x'' in (''a'' − ''r'', ''a'')&thinsp;∪&thinsp;(''a'', ''a'' + ''r'') we have {{nobr|''{{′|f}}''(''x'') > 0,}} then ''f'' is strictly increasing at ''a'' and has neither a local maximum nor a local minimum there.
* If none of the above conditions hold, then the test fails. (Such a condition is not [[vacuous truth|vacuous]]; there are functions that satisfy none of the first three conditions, e.g. ''f''(''x'') = ''x''<sup>2</sup>&thinsp;sin(1/''x'')).

Again, corresponding to the comments in the section on monotonicity properties, note that in the first two cases, the inequality is not required to be strict, while in the third, strict inequality is required.