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 which initial approximation will yield the closest second approximation to the solution $x = w$, we should analyze each option in the context of the graph provided.

1. Initial Approximation at $x_1 = a$: Starting at point $a$, we will apply Newton's method. Given that the graph seems to suggest a local extremum nearby, this choice may lead us away from $w$ after the first iteration.

2. Initial Approximation at $x_1 = b$: Choosing point $b$ is more favorable as it is closer to $w$. Since $b$ appears to be on the descending slope of the curve, it is expected that the subsequent approximation would converge more effectively towards the solution.

Thus, the better initial approximation is:

Answer: $x_1 = b$.