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- Feb 14, 2012
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Let $f(x)=x^3-3x+1$. Find the number of distinct real roots of the equation $f(f(x))=0$.
Yes, 7 is the correct answer!Just asking
I am getting Total no. of real solution is $ = 7$
Is is Right or not
Thanks
Finding the real roots of \(\displaystyle f(f(x)) = 0\) is equivalent to finding the ones of \(\displaystyle f(x) = x_i\) where \(\displaystyle x_i\) for i = 1, 2, 3 are the (real) roots of f(x).
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Solving for the number of real roots of \(\displaystyle f(x) = x_1=root_1\) means the same thing as finding the number of intersection points between the curve $y=f(x)$ and $y=root_1$, as shown by the green line and we see that in this case we have only one real root for \(\displaystyle f(x) = x_1=root_1\). | Solving for the number of real roots of \(\displaystyle f(x) = x_2=root_2\) means the same thing as finding the number of intersection points between the curve $y=f(x)$ and $y=root_2$, as shown by the purple line and we see that in this case we have only one real root for \(\displaystyle f(x) = x_2=root_2\). | Solving for the number of real roots of \(\displaystyle f(x) = x_3=root_3\) means the same thing as finding the number of intersection points between the curve $y=f(x)$ and $y=root_3$, as shown by the orange line and we see that in this case we have only one real root for \(\displaystyle f(x) = x_3=root_3\). |
May I see your approach on this problem?anemone said:this method is so much better than my first approach
May I see your approach on this problem? Balarka .
$f(x)=x^3-3x+1$ | $g(x)=f(f(x))=(f(x))^3-3(f(x))+1$ |
$f'(x)=3x^2-3=3(x^2-1)=3(x+1)(x-1)$ $\rightarrow f'(x)=0$ iff $3(x+1)(x-1)=0$ or $x=\pm1$. | $g'(x)=3f(x)^2f'(x)-3f'(x)=3f'(x)((f(x))^2-1)$ $\rightarrow g'(x)=0$ iff $x=\pm1$, $x=\pm \sqrt{3}$, or $x=-2$ or $x=0$. |
$f''(x)=6x$ | $g''(x)=6f(x)(f'(x))^2+3f''(x)((f(x))^2-1)$ |
$x$ | -2 | $-\sqrt{3}$ | -1 | 0 | 1 | $\sqrt{3}$ |
$f(x)$ | -1 | 1 | 3 | 1 | -1 | 1 |
$f'(x)$ | 9 | 6 | 0 | -3 | 0 | 6 |
$f''(x)$ | -12 | $-6\sqrt{3}$ | -6 | 0 | 6 | $6\sqrt{3}$ |
$g(x)$ | 3 | -1 | 19 | -1 | 3 | -1 |
$g'(x)$ | 0 | 0 | 0 | 0 | 0 | 0 |
$g''(x)$ | -486 | 216 | -144 | 54 | 0 | 216 |
Thank you for the compliment, Balarka!Hmm... very nice. I like the way you analyzed the behavior of f(f(x)), very well done. Mine may have been a little time consuming but this one is more representative.
PS I particularly like this solution.![]()