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First derivative test

107 bytes added, 21:14, 7 March 2013
What the test says: one-sided sign versions
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===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.
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===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.
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==Facts used==
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<center>{{#widget:YouTube|id=6PJplELQB1g}}</center> =Conclusive =Inconclusive and inconclusive conclusive cases==
===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]]
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===Conclusive cases===
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