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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$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3-3\mathrm{x}+1 \\ \mathrm{Note} \mathrm{that} \\ \mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right)=0 \mathrm{implies} \mathrm{that} \\ \mathrm{f}\left(\mathrm{x}\right)=\mathrm{c} \mathrm{where} \mathrm{c} \mathrm{is} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{f} \mathrm{i}.\mathrm{e}. \mathrm{f}\left(\mathrm{c}\right)=0 \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)=3{\mathrm{x}}^2-3=3\left(\mathrm{x}-1\right)\left(\mathrm{x}+1\right) \\ \mathrm{x}=-1 \mathrm{is} \mathrm{point} \mathrm{of} \mathrm{maximma} \mathrm{and} \mathrm{x}=1 \mathrm{is} \mathrm{point} \mathrm{of} \mathrm{minima}. \\ \mathrm{f}\left(-1\right)=-1+3+1=3 \\ \mathrm{f}\left(1\right)=1-3+1=-1. \\ \mathrm{Hence} \mathrm{rough} \mathrm{graph} \mathrm{will} \mathrm{be}: \end{gathered}$$


$$\begin{gathered} \mathrm{f}\left(-2\right)=-1 \\ \mathrm{f}\left(-1\right)=3 \\ \mathrm{f}\left(0\right)=1 \\ \mathrm{f}\left(1\right)=-1 \\ \mathrm{f}\left(2\right)=3 \\ \mathrm{Hence} \mathrm{one} \mathrm{root} \mathrm{lies} \mathrm{between} -2 \mathrm{and} -1. \\ \mathrm{other} \mathrm{lies} \mathrm{between} 0 \mathrm{and} 1 \mathrm{and} \mathrm{third} \mathrm{between} 1 \mathrm{and} 2. \\ \mathrm{If} \mathrm{c} \left(\mathrm{root}\right) \mathrm{lies} \mathrm{between} -2 \mathrm{and} -1 \\ \mathrm{Then} \mathrm{f}\left(\mathrm{x}\right)=\mathrm{c} \mathrm{will} \mathrm{give} \mathrm{one} \mathrm{root} \mathrm{because} \mathrm{line} \mathrm{y}=\mathrm{c} \mathrm{will} \mathrm{go} \mathrm{below} \\ \mathrm{y}=-1. \\ \mathrm{If} \mathrm{c} \left(\mathrm{root}\right) \mathrm{lies} \mathrm{between} 0 \mathrm{and} 1 \\ \mathrm{Then} \mathrm{f}\left(\mathrm{x}\right)=\mathrm{c} \mathrm{will} \mathrm{give} \mathrm{one} \mathrm{root} \mathrm{because} \mathrm{line} \mathrm{y}=\mathrm{c} \mathrm{will} \mathrm{cut} \mathrm{graph} \mathrm{at} \\ \mathrm{three} \mathrm{distinct} \mathrm{points} \mathrm{and} \mathrm{same} \mathrm{goes} \mathrm{for} \mathrm{root} \mathrm{between} 1 \mathrm{and} 2. \\ \mathrm{Hence} 7 \mathrm{real} \mathrm{roots}. \end{gathered}$$
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