Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 20259 by Tinkutara last updated on 24/Aug/17

Find the number of real roots of the equation f(x) = x^3 + 2x^2 + 2x + 1 = 0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:+\:\mathrm{2}{x}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:+\:\mathrm{1}\:=\:\mathrm{0} \\ $$

Answered by ajfour last updated on 24/Aug/17

f ′(x)=3x^2 +4x+2 >0 so just one real root.

$${f}\:'\left({x}\right)=\mathrm{3}{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{2}\:>\mathrm{0} \\ $$$${so}\:{just}\:\boldsymbol{{one}}\:\boldsymbol{{real}}\:\boldsymbol{{root}}. \\ $$

Commented by Tinkutara last updated on 25/Aug/17

Thank you very much Sir!

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$

Commented by Tinkutara last updated on 24/Aug/17

Why f′(x) > 0 ⇒ one real root only?

$$\mathrm{Why}\:{f}'\left({x}\right)\:>\:\mathrm{0}\:\Rightarrow\:\mathrm{one}\:\mathrm{real}\:\mathrm{root}\:\mathrm{only}? \\ $$

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 ′(x)=(dy/dx)=0 , if prior to this point f ′(x)<0 and after that f ′(x)>0 or first f ′(x)>0 then through zero it becomes <0 then it implies a turn is taken. line takes no turn has maximum one root (except y=0),a quadratic function takes one turn, max two roots, cubic function proceeds from ∓∞ to ±∞ .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 (in sketch in comment) .Anyhow if f (x) is cubic and f ′(x) never zero then it either keeps increasing (if coeff. of x^3 +ve) or keeps only decreasing (if coeff. of x^3 −ve)..

$${in}\:{order}\:{to}\:{cross}\:{x}-{axis}\:{again}\:{the} \\ $$$${curve}\:{needs}\:{to}\:{take}\:{a}\:{turn},\:{the} \\ $$$${point}\:{where}\:{tangent}\:{is}\:{horizontal} \\ $$$${f}\:'\left({x}\right)=\frac{{dy}}{{dx}}=\mathrm{0}\:,\:{if}\:{prior}\:{to}\:{this}\:{point}\: \\ $$$${f}\:'\left({x}\right)<\mathrm{0}\:{and}\:{after}\:{that}\:{f}\:'\left({x}\right)>\mathrm{0} \\ $$$${or}\:{first}\:{f}\:'\left({x}\right)>\mathrm{0}\:{then}\:{through} \\ $$$${zero}\:{it}\:{becomes}\:<\mathrm{0}\:{then}\:{it} \\ $$$${implies}\:{a}\:{turn}\:{is}\:{taken}.\:{line}\:{takes} \\ $$$${no}\:{turn}\:{has}\:{maximum}\:{one}\:{root} \\ $$$$\left({except}\:{y}=\mathrm{0}\right),{a}\:{quadratic} \\ $$$${function}\:{takes}\:{one}\:{turn},\:{max} \\ $$$${two}\:{roots},\: \\ $$$${cubic}\:{function}\:{proceeds}\:{from} \\ $$$$\mp\infty\:{to}\:\pm\infty\:.{in}\:{between}\:{if}\:{it} \\ $$$${does}\:{not}\:{take}\:{this}\:{turn}\:{it}\:{crosses} \\ $$$${x}\:{only}\:{once}.{And}\:{sometimes} \\ $$$${there}\:{is}\:{the}\:{turn}\:{but}\:{yet}\:{it}\:{crosses} \\ $$$${x}-{axis}\:{only}\:{once},\:{as}\:\left({in}\:{sketch}\:{in}\right. \\ $$$$\left.{comment}\right)\:.{Anyhow}\:{if}\:{f}\:\left({x}\right)\:{is} \\ $$$${cubic}\:{and}\:{f}\:'\left({x}\right)\:{never}\:{zero}\:{then}\:{it} \\ $$$${either}\:{keeps}\:{increasing}\:\left({if}\:{coeff}.\right. \\ $$$$\left.{of}\:{x}^{\mathrm{3}} \:+{ve}\right)\:{or}\:{keeps}\:{only} \\ $$$${decreasing}\:\left({if}\:{coeff}.\:{of}\:{x}^{\mathrm{3}} \:−{ve}\right).. \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com