Question Number 20259 by Tinkutara last updated on 24/Aug/17
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:+\:\mathrm{2}{x}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:+\:\mathrm{1}\:=\:\mathrm{0} \\ $$
Answered by ajfour last updated on 24/Aug/17
$${f}\:'\left({x}\right)=\mathrm{3}{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{2}\:>\mathrm{0} \\ $$$${so}\:{just}\:\boldsymbol{{one}}\:\boldsymbol{{real}}\:\boldsymbol{{root}}. \\ $$
Commented by Tinkutara last updated on 25/Aug/17
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$
Commented by Tinkutara last updated on 24/Aug/17
$$\mathrm{Why}\:{f}'\left({x}\right)\:>\:\mathrm{0}\:\Rightarrow\:\mathrm{one}\:\mathrm{real}\:\mathrm{root}\:\mathrm{only}? \\ $$
Commented by ajfour last updated on 24/Aug/17
Commented by ajfour last updated on 24/Aug/17
$${in}\:{order}\:{to}\:{cross}\:{x}-{axis}\:{again}\:{the} \\ $$$${curve}\:{needs}\:{to}\:{take}\:{a}\:{turn},\:{the} \\ $$$${point}\:{where}\:{tangent}\:{is}\:{horizontal} \\ $$$${f}\:'\left({x}\right)=\frac{{dy}}{{dx}}=\mathrm{0}\:,\:{if}\:{prior}\:{to}\:{this}\:{point}\: \\ $$$${f}\:'\left({x}\right)<\mathrm{0}\:{and}\:{after}\:{that}\:{f}\:'\left({x}\right)>\mathrm{0} \\ $$$${or}\:{first}\:{f}\:'\left({x}\right)>\mathrm{0}\:{then}\:{through} \\ $$$${zero}\:{it}\:{becomes}\:<\mathrm{0}\:{then}\:{it} \\ $$$${implies}\:{a}\:{turn}\:{is}\:{taken}.\:{line}\:{takes} \\ $$$${no}\:{turn}\:{has}\:{maximum}\:{one}\:{root} \\ $$$$\left({except}\:{y}=\mathrm{0}\right),{a}\:{quadratic} \\ $$$${function}\:{takes}\:{one}\:{turn},\:{max} \\ $$$${two}\:{roots},\: \\ $$$${cubic}\:{function}\:{proceeds}\:{from} \\ $$$$\mp\infty\:{to}\:\pm\infty\:.{in}\:{between}\:{if}\:{it} \\ $$$${does}\:{not}\:{take}\:{this}\:{turn}\:{it}\:{crosses} \\ $$$${x}\:{only}\:{once}.{And}\:{sometimes} \\ $$$${there}\:{is}\:{the}\:{turn}\:{but}\:{yet}\:{it}\:{crosses} \\ $$$${x}-{axis}\:{only}\:{once},\:{as}\:\left({in}\:{sketch}\:{in}\right. \\ $$$$\left.{comment}\right)\:.{Anyhow}\:{if}\:{f}\:\left({x}\right)\:{is} \\ $$$${cubic}\:{and}\:{f}\:'\left({x}\right)\:{never}\:{zero}\:{then}\:{it} \\ $$$${either}\:{keeps}\:{increasing}\:\left({if}\:{coeff}.\right. \\ $$$$\left.{of}\:{x}^{\mathrm{3}} \:+{ve}\right)\:{or}\:{keeps}\:{only} \\ $$$${decreasing}\:\left({if}\:{coeff}.\:{of}\:{x}^{\mathrm{3}} \:−{ve}\right).. \\ $$$$ \\ $$