91Ó°ÊÓ

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$$\begin{gathered} 1.\mathit{F}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}\mathit{l}\mathit{y}\mathbf{}\mathit{D}\mathit{e}\mathit{f}\mathit{i}\mathit{n}\mathit{i}\mathit{n}\mathit{g}\mathbf{}\mathit{C}\mathit{o}\mathit{n}\mathit{c}\mathit{a}\mathit{v}\mathit{i}\mathit{t}\mathit{y}\mathbf{:} \\ Supposefandf\text{'}arebothdifferentiableonanintervalI. \\ \left(a\right)fisconcaveuponIiff\text{'}isincrea\sin gonI. \\ \left(b\right)fisconcavedownonIiff\text{'}isdecrea\sin gonI. \end{gathered}$$

$$\begin{gathered} 2.\mathit{T}\mathit{h}\mathit{e}\mathbf{}\mathit{S}\mathit{e}\mathit{c}\mathit{o}\mathit{n}\mathit{d}\mathbf{}\mathit{D}\mathit{e}\mathit{r}\mathit{i}\mathit{v}\mathit{a}\mathit{t}\mathit{i}\mathit{v}\mathit{e}\mathbf{}\mathit{D}\mathit{e}\mathit{t}\mathit{e}\mathit{r}\mathit{m}\mathit{i}\mathit{n}\mathit{e}\mathit{s}\mathbf{}\mathit{C}\mathit{o}\mathit{n}\mathit{c}\mathit{a}\mathit{v}\mathit{i}\mathit{t}\mathit{y}\mathbf{:} \\ Supposebothfandf\text{'}aredifferentiableonanintervalI. \\ \left(a\right)Iff\text{'}\text{'}ispositiveonI,thenfisconcaveuponI. \\ \left(b\right)Iff\text{'}\text{'}isnegativeonI,thenfisconcavedownonI. \end{gathered}$$

role="math" localid="1649778521794" $$\begin{gathered} 3.\mathit{T}\mathit{h}\mathit{e}\mathbf{}\mathit{S}\mathit{e}\mathit{c}\mathit{o}\mathit{n}\mathit{d}\mathbf{-}\mathit{D}\mathit{e}\mathit{r}\mathit{i}\mathit{v}\mathit{a}\mathit{t}\mathit{i}\mathit{v}\mathit{e}\mathbf{}\mathit{T}\mathit{e}\mathit{s}\mathit{t}\mathbf{:} \\ Supposex=cisthelocationofacriticalpointofafunctionfwithf\text{'}\left(c\right)=0, \\ andsupposebothfandf\text{'}aredifferentiableandf\text{'}\text{'}iscontinuousonaninterval \\ aroundx=c. \\ \left(a\right)Iff\text{'}\text{'}\left(c\right)ispositive,thenfhasalocalminimumatx=c. \\ \left(b\right)Iff\text{'}\text{'}\left(c\right)isnegative,thenfhasalocalmaximumatx=c. \\ \left(c\right)Iff\text{'}\text{'}\left(c\right)=0,thenthistestsaysnothingaboutwhetherornotfhasanextremum \\ atx=c. \end{gathered}$$