Let f(x) = x^3 - 3x + 1. Find the number of different real solution of the equation f(f(x)) = 0

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Please find below the solution to the asked query:

fx=x3-3x+1Note thatffx=0 implies thatfx=c where c is the roots of f i.e. fc=0f'x=3x2-3=3x-1x+1x=-1 is point of maximma and x=1 is point of minima.f-1=-1+3+1=3f1=1-3+1=-1. Hence rough graph will be:

f-2=-1f-1=3f0=1f1=-1f2=3Hence one root lies between -2 and -1.other lies between 0 and 1 and third between  1 and 2.If c root lies between -2 and -1Then fx=c will give one root because line y=c will go belowy=-1. If c root lies between 0 and 1Then fx=c will give one root because line y=c will cut graph atthree distinct points and same goes for root between 1 and 2.Hence 7 real roots.
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