Newton’s method can be used to give an approximation close to the solution $x = w$ - HSC - SSCE Mathematics Extension 1 - Question 9 - 2018 - Paper 1

To determine the best initial approximation $x_1$, we analyze the graph shown in the question.

From the graph, it is evident that the function approaches the solution $x = w$ more closely when starting from the value of $b$ compared to $a$. As Newton's method involves considering the slope of the tangent at the approximation point, choosing $x_1 = b$ allows the tangent line to intersect the x-axis closer to the point $w$, ultimately yielding a second approximation that is nearer to the actual solution.

Thus, the correct choice is:

Answer: $x_1 = b$.