Editing Derivative test (section)

==First-derivative test==
The first-derivative test examines a function's [[monotonic function|monotonic]] properties (where the function is increasing or decreasing), focusing on a particular point in its [[domain of a function|domain]]. If the function "switches" from increasing to decreasing at the point, then the function will achieve a highest value at that point. Similarly, if the function "switches" from decreasing to increasing at the point, then it will achieve a least value at that point. If the function fails to "switch" and remains increasing or remains decreasing, then no highest or least value is achieved.

One can examine a function's monotonicity without calculus. However, calculus is usually helpful because there are [[necessary and sufficient conditions|sufficient conditions]] that guarantee the monotonicity properties above, and these conditions apply to the vast majority of functions one would encounter.

===Precise statement of monotonicity properties===
Stated precisely, suppose that ''f'' is a [[real number|real]]-valued function defined on some [[open interval]] containing the point ''x'' and suppose further that ''f'' is [[Continuous function|continuous]] at ''x''.

* If there exists a positive number ''r''&nbsp;>&thinsp;0 such that ''f'' is weakly increasing on {{open-closed|''x'' − ''r'', ''x''}} and weakly decreasing on {{closed-open|''x'', ''x'' + ''r''}}, then ''f'' has a local maximum at ''x''.
* If there exists a positive number ''r''&nbsp;>&thinsp;0 such that ''f'' is strictly increasing on {{open-closed|''x'' − ''r'', ''x''}} and strictly increasing on {{closed-open|''x'', ''x'' + ''r''}}, then ''f'' is strictly increasing on {{open-open|''x'' − ''r'', ''x'' + ''r''}} and does not have a local maximum or minimum at ''x''.
Note that in the first case, ''f'' is not required to be strictly increasing or strictly decreasing to the left or right of ''x'', while in the last case, ''f'' is required to be strictly increasing or strictly decreasing. The reason is that in the definition of local maximum and minimum, the inequality is not required to be strict: e.g. every value of a [[constant function]] is considered both a local maximum and a local minimum.

===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.

===Applications===

The first-derivative test is helpful in solving [[optimization problem]]s in physics, economics, and engineering. In conjunction with the [[extreme value theorem]], it can be used to find the absolute maximum and minimum of a real-valued function defined on a [[closed interval|closed]] and [[bounded set|bounded]] interval. In conjunction with other information such as concavity, inflection points, and [[asymptote]]s, it can be used to sketch the [[graph of a function|graph]] of a function.