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- Apr 14, 2013

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The function \begin{equation*}f(x)=\frac{x}{e^{x/9}}\cdot \frac{\sin \left (\pi (x-1)\right )}{x-1}\end{equation*} has at exactly one position $\overline{x}>1$ the same value as at the position $x=1$. Determine the position $\overline{x}$ using an iteration method with accuracy of two decimal digits.

I have done the following:

We have that \begin{equation*}f(\overline{x})=f(1)\Rightarrow f(\overline{x})-f(1)=0 \Rightarrow g(x):=f(x)-f(1)\end{equation*}

First we have to calculate $f(1)$:

\begin{align*}f(1)&=\lim_{x\rightarrow 1}f(x)=\lim_{x\rightarrow 1}\frac{x}{e^{x/9}}\cdot \frac{\sin \left (\pi (x-1)\right )}{x-1}=\frac{1}{e^{1/9}}\cdot \lim_{x\rightarrow 1}\frac{\sin \left (\pi (x-1)\right )}{x-1}\ \overset{DLH}{ = } \ \frac{1}{e^{1/9}}\cdot \lim_{x\rightarrow 1}\frac{\pi \cos \left (\pi (x-1)\right )}{1}\\ & =\frac{1}{e^{1/9}}\cdot \pi \cos \left (\pi \cdot 0\right )=\frac{1}{e^{1/9}}\cdot \pi =\frac{\pi}{e^{1/9}}\end{align*}

Therefore we get the function \begin{equation*}g(x)=\frac{x}{e^{x/9}}\cdot \frac{\sin \left (\pi (x-1)\right )}{x-1}-\frac{\pi}{e^{1/9}}\end{equation*}

Now we have to apply an iteration method to approximate the root of that function, right? Do we use the Newton's method?

We don't have an interval to which the root will belong, we only know that it is greater than $1$. So do we have to guess such an interval to calculate the first input $x_0$?