Determine the number of iteration to find solutions accurate to within $10^{-2}$ for $f(x)=x^3-7x^2+14x-6=0$ on $[a,b]=[1,3.2]$

Question

i got the number of iteration,$n$, to achieve the accuracy, $\epsilon=10^{-2}$ is $n=5.5\approx 6$

But in answer script, $n=8$.

My procedure is

$ \frac{(b-a)}{2^n}<\epsilon$

$\Rightarrow\frac{(3.2-1)}{2^{n}}<10^{-2}$

$\Rightarrow{(2.2)}{2^{-n}}<10^{-2}$

Taking $\log_{10}$

$\log_{10}(2.2)-n\log_{10}(2)<-2$

$\Rightarrow n>\frac{2-n\log_{10}(2)}{\log_{10}(2.2)}=5.5\approx 6$

Where is the mistake?

Answer 1

You want $\frac{2.2}{2^n} < 10^{-2}$, or in other words, $220 < 2^n$. Then $\log_2 220 < n$. Since $\log_2 220 \approx 7.7814\cdots$, we see that the smallest such $n$ is $n=8$.

Alternatively, you could notice that $f(3) = 0$., and so $n=0$ :-).