To apply Newton's method for this problem, we follow these steps:

Newton's method formula: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Analysis and Step-by-Step Solution:

1. Initial Approximation:

   • The problem states that the initial approximation $x_1 = 6$.

2. Tangent at $x_1 = 6$:

   • From the graph, observe the value of $f(x_1)$ (the y-coordinate of the function at $x = 6$) and estimate $f'(x_1)$ by examining the slope of the tangent at $x = 6$.
   • Draw the tangent line at $x = 6$ on the graph and determine where it crosses the x-axis. This intersection is the next approximation $x_2$.

3. Repeating for $x_2$:

   • Once $x_2$ is estimated, use the same process: observe the y-coordinate $f(x_2)$, approximate the slope $f'(x_2)$, and determine where the tangent line at $x_2$ intersects the x-axis.
   • This intersection will give the next approximation $x_3$.

4. Estimations:

   • Since this problem is graphical, we’ll use visual approximations for $f(x)$ and $f'(x)$ based on the tangent lines drawn.

Please let me know if you would like further clarification or assistance with estimating values directly from the graph, as I can guide you through interpreting the points and tangents.

Related Questions:

1. How does Newton's method converge to the root of a function?
2. What are some conditions where Newton's method might fail to converge?
3. Can Newton's method be used if the derivative $f'(x)$ is zero at any point?
4. How do we find the derivative $f'(x)$ graphically?
5. What are some alternatives to Newton's method for root-finding?

Tip: Newton’s method can be very fast if the initial guess is close to the actual root, but it may diverge if the initial guess is too far.