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First derivative test

86 bytes added, 23:24, 3 May 2012
What the test says: one-sided sign versions
! Continuity and differentiability assumption !!Hypothesis on sign of derivative!! Conclusion !! Prototypical pictures (the dotted point corresponds to <math>(c,f(c))</math>, and the dashed line is the one-sided tangent line at the point)
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| <math>f</math> is left continuous ''at'' <math>c</math> and differentiable on the immediate left of <math>c</math> || <math>\! f'(x)</math> is positive (respectively, nonnegative) for <math>x</math> to the immediate left of <math>c</math> (i.e., for <math>x \in (c - \delta, c)</math> for sufficiently small <math>\delta > 0</math>)|| <math>f</math> has a strict local maximum from the left at <math>c</math>, i.e., <math>f(c) > f(x)</math> (respectively, <math>f</math> has a local maximum from the left at <math>c</math>, i.e., <math>f(c) \ge f(x)</math>) for <math>x</math> to the immediate left of <math>c</math>. || [[File:Leftincreasingconcaveup.png|100px]][[File:Leftincreasingconcavedownflat.png|100px80px]]
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|<math>f</math> is left continuous ''at'' <math>c</math> and differentiable on the immediate left of <math>c</math> ||<math>\! f'(x)</math> is negative (respectively, nonpositive) for <math>x</math> to the immediate left of <math>c</math> (i.e., for <math>x \in (c - \delta, c)</math> for sufficiently small <math>\delta > 0</math>)|| <math>f</math> has a strict local minimum from the left at <math>c</math>, i.e., <math>f(c) < f(x)</math> (respectively, <math>f</math> has a local minimum from the left at <math>c</math>, i.e., <math>f(c) \le f(x)</math>) for <math>x</math> to the immediate left of <math>c</math>. || [[File:Leftdecreasingconcaveupflat.png|100px80px]][[File:Leftdecreasingconcavedown.png|100px]]
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| <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 positive (respectively, nonnegative) 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 minimum from the right at <math>c</math>, i.e., <math>f(c) < f(x)</math> (respectively, <math>f</math> has a local minimum from the right at <math>c</math>, i.e., <math>f(c) \le f(x)</math>) for <math>x</math> to the immediate right of <math>c</math>. || [[File:Rightincreasingconcaveupnotflat.png|100px]][[File:Rightincreasingconcavedown.png|100px]]
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| <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|100px]]
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