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Question Number 33658 by rahul 19 last updated on 21/Apr/18
$$\boldsymbol{{I}}{f}\:{f}\left({x}\right)=\:{x}^{\mathrm{3}} −\mathrm{3}{x}+\mathrm{1} \\ $$$${then}\:{find}\:{number}\:{of}\:{different}\:\:\:{real} \\ $$$${solutions}\:{of}\:{f}\left({f}\left({x}\right)\right)=\mathrm{0}\:? \\ $$
Commented by rahul 19 last updated on 21/Apr/18
$${pls}\:{help}. \\ $$
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](Q33668.png)
$${f}\left({x}\right)=\mathrm{0} \\ $$$${x}_{\mathrm{1}} =−\mathrm{2cos}\:\frac{\pi}{\mathrm{9}}\approx−\mathrm{1}.\mathrm{879} \\ $$$${x}_{\mathrm{2}} =\mathrm{2sin}\:\frac{\pi}{\mathrm{18}}\approx\mathrm{0}.\mathrm{347} \\ $$$${x}_{\mathrm{3}} =\mathrm{2cos}\:\frac{\mathrm{2}\pi}{\mathrm{9}}\approx\mathrm{1}.\mathrm{532} \\ $$$$\:\:\:\:\:\:\:\:\:\:\left[\mathrm{I}\:\mathrm{can}\:\mathrm{show}\:\mathrm{the}\:\mathrm{way}\:\mathrm{to}\:\mathrm{the}\:\mathrm{exact}\:\mathrm{solution}\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\mathrm{but}\:\mathrm{you}\:\mathrm{can}\:\mathrm{also}\:\mathrm{solve}\:\mathrm{by}\:\mathrm{try}\:\&\:\mathrm{error}\right] \\ $$$$ \\ $$$$\left.\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{f}\left({x}\right)=\right]−\infty;\infty\left[\:\mathrm{but}\:\mathrm{the}\:\mathrm{function}\right. \\ $$$$\mathrm{reaches}\:\mathrm{some}\:\mathrm{values}\:\mathrm{2}\:\mathrm{or}\:\mathrm{3}\:\mathrm{times}.\:\mathrm{so}\:\mathrm{we}\:\mathrm{need} \\ $$$$\mathrm{the}\:\mathrm{local}\:\mathrm{minimum}\:\mathrm{and}\:\mathrm{maximum},\:\mathrm{at}\:{f}'\left({x}\right)=\mathrm{0} \\ $$$$\mathrm{min}\left({f}\left({x}\right)\right)\in\left[{x}_{\mathrm{2}} ;\:{x}_{\mathrm{3}} \right] \\ $$$$\mathrm{max}\left({f}\left({x}\right)\right)\in\left[{x}_{\mathrm{1}} ;\:{x}_{\mathrm{2}} \right] \\ $$$${f}'\left({x}\right)=\mathrm{3}{x}^{\mathrm{2}} −\mathrm{3}=\mathrm{0}\:\Rightarrow\:\mathrm{Min}=\begin{pmatrix}{\mathrm{1}}\\{−\mathrm{1}}\end{pmatrix};\:\mathrm{Max}=\begin{pmatrix}{−\mathrm{1}}\\{\mathrm{3}}\end{pmatrix} \\ $$$$ \\ $$$$\left.\mathrm{so}\:{y}={f}\left({x}\right)\:\mathrm{reaches}\:{y}\in\:\right]−\infty;−\mathrm{1}\left[\:\cap\:\right]\mathrm{3};\infty\left[\:\mathrm{once},\right. \\ $$$$\left.{y}=−\mathrm{1}\wedge{y}=\mathrm{3}\:\mathrm{twice}\:\mathrm{and}\:{y}\in\:\right]−\mathrm{1};\mathrm{3}\left[\:\mathrm{three}\:\mathrm{times}\right. \\ $$$$ \\ $$$$\Rightarrow\:{f}\left({x}\right)\:\mathrm{reaches}\:{y}={x}_{\mathrm{1}} \:\mathrm{once},\:{y}={x}_{\mathrm{2}} \:\mathrm{three}\:\mathrm{times} \\ $$$$\mathrm{and}\:{y}={x}_{\mathrm{3}} \:\mathrm{three}\:\mathrm{times}\:\Rightarrow\:{f}\left({f}\left({x}\right)\right)\:\mathrm{has}\:\mathrm{7}\:\mathrm{real}\:\mathrm{zeros} \\ $$
Commented by rahul 19 last updated on 22/Apr/18
$${thank}\:{u}\:{so}\:{much}\:{sir}! \\ $$