→What the test says: one-sided sign versions
===What the test says: combined sign versions===
Note that if <math>f'</math> has ambiguous sign on the immediate left or on the immediate right of <math>c</math>, the first derivative test is inconclusive.
===Relation with critical points===
* In general, if the derivative changes sign as we move from the immediate left of the point to the immediate right of the point, then there is a local extremum at the point. If the derivative has the same sign on the immediate left and immediate right, we ''do not'' get a local extremum at the point.
Conclusive and inconclusive cases==
| The derivative of the function has oscillatory (ambiguous) sign on the immediate left and/or immediate right of the point || We cannot do sign analysis on the derivative on the immediate left and/or immediate right. Thus, it will not be possible to apply the first derivative test. All the possibilities (local maximum, local minimum, neither) remain open. || [[First derivative test is inconclusive for function whose derivative has ambiguous sign around the point]] || || [[File:Firstderivativetestfails.png|200px]]