I must use the Newton Raphson method.
Can someone please explain the steps I must take in order to reach the answer. For alot of this im having to self study the material to catch up and without instruction, im quite lost on how to approach the problem. I wasnt given any initial values, so am i to assume that i must use the distance formula or..
The closest point of the curve has an $x$-coordinate that is the solution of the equation $$f(x)=x^2+\log x =0$$ as $f\left(\frac12\right)=\frac14-\log 2<0$ and $f(1)=1>0$, for the intermediate values theorem the root is in the interval $(0.5,1)$
So we take $x_0=0.5$ and $x_n=x_{n-1}-\frac{f(x_{n-1})}{f'(x_{n-1})}$
$$ \begin{array}{l|r|r} n & x_n & error\\ \hline 0 & 0.5 &\\ 1 & 0.647716 & 0.147716 \\ 2 & 0.652917 & 0.00520115 \\ 3 & 0.652919 & 1.77\times 10^{-6} \\ 4 & 0.6529186404 & 1.9\times 10^{-13} \\ \end{array} $$
The solution with $6$ exact decimal places is $x^*=0.652919$