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First derivative test

No change in size, 00:07, 4 May 2012
What the test says: combined sign versions
! Continuity and differentiability assumption !! Sign of the derivative <math>f'</math> on immediate left of <math>c</math> !! Sign of <math>f'</math> on immediate right of <math>c</math> !! Conclusion for <math>f</math> at <math>c</math>: local minimum, local maximum, or neither? !! Prototypical pictures (the dotted point is <math>(c,f(c))</math>)
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| <math>f</math> is continuous at <math>c</math> and differentiable on the immediate left and immediate right of <math>c</math> || positive || negative || strict local maximum (two-sided) || [[File:StrictlocalmaxwithzeroderivativeStrictlocalmaxwithderivativezero.png|100px]] [[File:Strictlocalmaxwithundefinedderivative.png|100px]]
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| <math>f</math> is continuous at <math>c</math> and differentiable on the immediate left and immediate right of <math>c</math> || negative || positive || strict local minimum (two-sided) || [[File:StrictlocalminwithzeroderivativeStrictlocalminwithderivativezero.png|100px]] [[File:Strictlocalminwithundefinedderivative.png|100px]]
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| <math>f</math> is continuous at <math>c</math> and differentiable on the immediate left and immediate right of <math>c</math> || positive || positive || neither local maximum nor local minimum, because <math>f</math> is increasing through the point ||
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