Differentiation Questions

Question Number 100388 by Ar Brandon last updated on 26/Jun/20

$$\:\:\:\:\:\:\:\mathcal{G}\mathrm{iven}\:\mathrm{f}:\left[\mathrm{0},\mathrm{2}\right]\rightarrow\mathbb{R}\:,\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{twice}\:\mathrm{derivable}\:\mathrm{and}\: \\$$$$\mathrm{f}\left(\mathrm{0}\right)=\mathrm{f}\left(\mathrm{1}\right)=\mathrm{f}\left(\mathrm{2}\right)=\mathrm{0} \\$$$${i}-\mathcal{S}\mathrm{how}\:\mathrm{that}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{c}_{\mathrm{1}} ,\:\mathrm{c}_{\mathrm{2}} ,\:\mathrm{such}\:\mathrm{that}\:\mathrm{f}'\left(\mathrm{c}_{\mathrm{1}} \right)=\mathrm{0}\: \\$$$$\mathrm{and}\:\mathrm{f}'\left(\mathrm{c}_{\mathrm{2}} \right)=\mathrm{0} \\$$$${ii}-\mathcal{S}\mathrm{how}\:\mathrm{that}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{c}_{\mathrm{3}} \:\mathrm{such}\:\mathrm{that}\:\mathrm{f}''\left(\mathrm{c}_{\mathrm{3}} \right)=\mathrm{0} \\$$

Answered by maths mind last updated on 26/Jun/20

$$\left.{f}\left(\mathrm{0}\right)={f}\left(\mathrm{1}\right)\Rightarrow\exists{c}\in\right]\mathrm{0},\mathrm{1}\left[,\:\:{f}'\left({c}_{\mathrm{1}} \right)=\mathrm{0}\right. \\$$$$\left.{f}\left(\mathrm{1}\right)={f}\left(\mathrm{2}\right)\Rightarrow\exists{c}_{\mathrm{2}} \in\right]\mathrm{1},\mathrm{2}\left[\:{f}'\left({c}_{\mathrm{2}} \right)=\mathrm{0}\right. \\$$$${let}\:{f}'\left({x}\right)\:{over}\:\left[{c}_{\mathrm{1}} ,{c}_{\mathrm{2}} \right]\: \\$$$${f}'\left({c}_{\mathrm{1}} \right)={f}'\left({c}_{\mathrm{2}} \right)\Rightarrow\exists{c}_{\mathrm{3}} \in\left[{c}_{\mathrm{1}} ,{c}_{\mathrm{2}} \right]{such}\:{f}''\left({c}_{\mathrm{3}} \right)=\mathrm{0} \\$$$${i}\:{used}\:{if}\:{f}\:{continus}\:{differentiabl}\:{over}\:\left[{a},{b}\right]\:{such} \\$$$${f}\left({a}\right)={f}\left({b}\right)\Rightarrow\exists{c}\in\left[{a},{b}\right]\:{such}\:{f}'\left({c}\right)=\mathrm{0}.{Roll}\:{theorem} \\$$$$\\$$$$\\$$

Commented by DGmichael last updated on 26/Jun/20

��very good.

Commented by Ar Brandon last updated on 26/Jun/20

Thanks ��

Commented by Ar Brandon last updated on 26/Jun/20

Ouaye DG, dès que ce monsieur se connecte oulala !�� On dirait une machine qui venait d'être activée.����