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If-f-x-x-3-3x-1-then-find-number-of-different-real-solutions-of-f-f-x-0-




Question Number 33658 by rahul 19 last updated on 21/Apr/18
If f(x)= x^3 −3x+1  then find number of different   real  solutions of f(f(x))=0 ?
\boldsymbolIff(x)=x33x+1thenfindnumberofdifferentrealsolutionsoff(f(x))=0?
Commented by rahul 19 last updated on 21/Apr/18
pls help.
plshelp.
Answered by MJS last updated on 21/Apr/18
f(x)=0  x_1 =−2cos (π/9)≈−1.879  x_2 =2sin (π/(18))≈0.347  x_3 =2cos ((2π)/9)≈1.532            [I can show the way to the exact solution             but you can also solve by try & error]    the range of f(x)=]−∞;∞[ but the function  reaches some values 2 or 3 times. so we need  the local minimum and maximum, at f′(x)=0  min(f(x))∈[x_2 ; x_3 ]  max(f(x))∈[x_1 ; x_2 ]  f′(x)=3x^2 −3=0 ⇒ Min= ((1),((−1)) ); Max= (((−1)),(3) )    so y=f(x) reaches y∈ ]−∞;−1[ ∩ ]3;∞[ once,  y=−1∧y=3 twice and y∈ ]−1;3[ three times    ⇒ f(x) reaches y=x_1  once, y=x_2  three times  and y=x_3  three times ⇒ f(f(x)) has 7 real zeros
f(x)=0x1=2cosπ91.879x2=2sinπ180.347x3=2cos2π91.532[Icanshowthewaytotheexactsolutionbutyoucanalsosolvebytry&error]therangeoff(x)=];[butthefunctionreachessomevalues2or3times.soweneedthelocalminimumandmaximum,atf(x)=0min(f(x))[x2;x3]max(f(x))[x1;x2]f(x)=3x23=0Min=(11);Max=(13)soy=f(x)reachesy];1[]3;[once,y=1y=3twiceandy]1;3[threetimesf(x)reachesy=x1once,y=x2threetimesandy=x3threetimesf(f(x))has7realzeros
Commented by rahul 19 last updated on 22/Apr/18
thank u so much sir!
thankusomuchsir!thankusomuchsir!

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