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First derivative test

214 bytes added, 21:14, 7 March 2013
What the test says: one-sided sign versions
| <math>f</math> is right continuous ''at'' <math>c</math> and differentiable on the immediate right of <math>c</math> ||<math>\! f'(x)</math> is negative (respectively, nonpositive) for <math>x</math> to the immediate right of <math>c</math> (i.e., for <math>x \in (c,c + \delta)</math> for sufficiently small <math>\delta > 0</math>)|| <math>f</math> has a strict local maximum from the right at <math>c</math>, i.e., <math>f(c) > f(x)</math> (respectively, <math>f</math> has a local maximum from the right at <math>c</math>, i.e., <math>f(c) \ge f(x)</math>) for <math>x</math> to the immediate right of <math>c</math>. || [[File:Rightdecreasingconcavedownnotflat.png|100px]][[File:Rightdecreasingconcaveup.png|80px]]
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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===
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! What problem do we run into? !! What kind of trouble can we have? !! Link to example !! Can this be fixed? Remedy that may work !! Picture
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| The function is not continuous at the critical point || We may be able to do sign analysis of the derivative on the immediate left and immediate right, but draw incorrect conclusions by applying the one-sided or combined sign version of the first derivative test. A priori, all the possibilities (local maximum, local minimum, neither) remain open. || [[first derivative test fails for function that is discontinuous at the critical point]] || If the function has one-sided limits at the critical point: [[variation of first derivative test for discontinuous function with one-sided limits]] ||
| 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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