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Question Number 99430 by I want to learn more last updated on 20/Jun/20

Answered by mr W last updated on 20/Jun/20

$$f\left(x\right)=x^3+\frac{1}{6}{x}^2-\frac{44}{9}x-\frac{40}{9}=0$$$$f{}^{\prime }\left(x\right)=3x^2+\frac{1}{3}x-\frac{44}{9}=0$$$$\Rightarrow 27x^2+3x-44=0$$$$\Rightarrow x=-\frac{4}{3},\frac{11}{9}$$$$oneofthemisadoublerootoff\left(x\right)=0.$$$$aftercheckingweknowitisx=-\frac{4}{3}.$$$$saytheotherrootis\beta ,then$$$$x^3+\frac{1}{6}{x}^2-\frac{44}{9}x-\frac{40}{9}={(x+\frac{4}{3})}^2(x-\beta )$$$$withx=0weget$$$$-\frac{40}{9}={\left(\frac{4}{3}\right)}^2(-\beta )$$$$\Rightarrow \beta =\frac{5}{2}$$$$\Rightarrow therootsarex=-\frac{4}{3},-\frac{4}{3}and\frac{5}{2}$$

Commented by floor(10²Eta[1]) last updated on 21/Jun/20

$$$$$$$$$$okbut{what}^{\prime }stheproof$$

Commented by I want to learn more last updated on 20/Jun/20

$$\mathrm{Thanks}\mathrm{sir},\mathrm{i}\mathrm{appreciate}.$$

Commented by floor(10²Eta[1]) last updated on 21/Jun/20

$$$$$$$$$$canyouexplainmewhydoing{f}^{\prime }\left(x\right)=0$$$$youknowthatoneoftherootsarerepeated\int$$

Commented by 1549442205 last updated on 21/Jun/20

$$\mathrm{we}\mathrm{have}\mathrm{the}\mathrm{following}\mathrm{property}:$$$$\mathrm{if}\mathrm{the}\mathrm{equation}\mathrm{f}\left(\mathrm{x}\right)=0\mathrm{has}\mathrm{one}\mathrm{double}\mathrm{root}\mathrm{then}$$$$\mathrm{that}\mathrm{root}\mathrm{is}\mathrm{a}\mathrm{root}\mathrm{of}\mathrm{the}\mathrm{equation}\mathrm{f}{}^{\prime }\left(\mathrm{x}\right)=0$$

Commented by mr W last updated on 21/Jun/20

Commented by mr W last updated on 21/Jun/20

$$iftherearethreerealroots,Pand$$$$Qmustlieaboveandunderthe$$$$x-axis.$$

Commented by PRITHWISH SEN 2 last updated on 21/Jun/20

$$\mathrm{let}\mathbf{a}\mathrm{and}\mathbf{b}\mathrm{be}\mathrm{the}\mathrm{roots}\mathrm{of}\mathrm{f}\left(\mathrm{x}\right)\mathrm{then}$$$$\mathrm{f}\left(\mathrm{x}\right)={(\mathrm{x}-\mathrm{a})}^2(\mathrm{x}-\mathrm{b})$$$${\mathbf{f}}^{\prime }\left(\mathrm{x}\right)=2(\mathrm{x}-\mathrm{a})(\mathrm{x}-\mathrm{b})+{(\mathrm{x}-\mathrm{a})}^2$$$$\mathrm{it}\mathrm{is}\mathrm{clear}\mathrm{that}\mathrm{a}\mathrm{is}\mathrm{also}\mathrm{the}\mathrm{root}\mathrm{of}{\mathbf{f}}^{\prime }\left(\mathrm{x}\right).\mathbf{proof}$$

Commented by mr W last updated on 21/Jun/20

Commented by mr W last updated on 21/Jun/20

$$ifthereisadoubleroot,onefrom$$$$PandQmustlieonthex-axis.$$

Commented by mr W last updated on 21/Jun/20

Commented by mr W last updated on 21/Jun/20

$$ifthereisonlyonerealroot,then$$$$PandQ{don}^{\prime }texistorbothofthem$$$$mustlieaboveorunderthex-axis.$$

Commented by mr W last updated on 21/Jun/20

$$adoublerootmeans:thecurvef\left(x\right)$$$$tangentsthex-axis.$$

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