← MATH 01130 · Calculus I
MATH 01130 · Applications of derivatives
First/second derivative tests
Definition
First derivative test: if f′ changes − to + at c, local min; + to −, local max. Second derivative test: if f′(c)=0 and f′′(c)>0, local min; f′′(c)<0, local max; f′′(c)=0, inconclusive.
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Prefer first derivative test when f′′ is messy or zero.
Critical points are where f′=0 or f′ DNE (in the domain). Endpoints need separate checking for absolute extrema on closed intervals.
Always verify the point is in the domain and that a sign change (or conclusive f′′) actually happens.
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1.Classify critical points of f(x)=x⁴−2x².