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Question Number 108422 by maouame last updated on 16/Aug/20

Answered by bubugne last updated on 17/Aug/20

$$$$$$1-solutiontriviale$$$$\mathrm{\exists }ϵ\in {\mathbb{R}}_{+}^{\ast }\mathrm{\forall }c\in ]0,ϵ]f\left(c\right)=f{}^{\prime }\left(c\right)=0$$$$$$$$2-solutionnontriviale$$$$f\left(0\right)=0etf{}^{\prime }\left(0\right)=0\Rightarrow \mathrm{\exists }ϵ\in {\mathbb{R}}_{+}^{\ast }\mathrm{\forall }x\in ]0,ϵ]f{}^{\prime }\left(x\right)\ne 0etf\left(x\right)\times f{}^{\prime }\left(x\right)>0$$$$\begin{array}{c}\mathrm{\exists }ϵ\in {\mathbb{R}}_{+}^{\ast }\mathrm{\forall }x\in ]0,ϵ]f\left(x\right)\times f{}^{\prime }\left(x\right)>0\\ \mathrm{\exists }a\in {\mathbb{R}}_{+}^{\ast }f\left(a\right)\times f{}^{\prime }\left(a\right)<0\end{array}\}\Rightarrow \mathrm{\exists }c\in ]ϵ,a[f{}^{\prime }\left(c\right)=0$$$$$$

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