Changes

First derivative test

, 00:07, 4 May 2012
What the test says: combined sign versions
! Continuity and differentiability assumption !! Sign of the derivative [itex]f'[/itex] on immediate left of [itex]c[/itex] !! Sign of [itex]f'[/itex] on immediate right of [itex]c[/itex] !! Conclusion for [itex]f[/itex] at [itex]c[/itex]: local minimum, local maximum, or neither? !! Prototypical pictures (the dotted point is [itex](c,f(c))[/itex])
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| [itex]f[/itex] is continuous at [itex]c[/itex] and differentiable on the immediate left and immediate right of [itex]c[/itex] || positive || negative || strict local maximum (two-sided) || [[File:StrictlocalmaxwithzeroderivativeStrictlocalmaxwithderivativezero.png|100px]] [[File:Strictlocalmaxwithundefinedderivative.png|100px]]
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| [itex]f[/itex] is continuous at [itex]c[/itex] and differentiable on the immediate left and immediate right of [itex]c[/itex] || negative || positive || strict local minimum (two-sided) || [[File:StrictlocalminwithzeroderivativeStrictlocalminwithderivativezero.png|100px]] [[File:Strictlocalminwithundefinedderivative.png|100px]]
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| [itex]f[/itex] is continuous at [itex]c[/itex] and differentiable on the immediate left and immediate right of [itex]c[/itex] || positive || positive || neither local maximum nor local minimum, because [itex]f[/itex] is increasing through the point ||
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