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 gives the closest second approximation to the solution $x = w$, we must consider the nature of Newton's method. This iterative process uses the function's derivative to progressively approximate the root.

• If we choose $x_1 = a$, we can observe the behavior of the function near point $a$. Given that the function is concave and the second approximation will depend heavily on the value of the derivative at $x_1$.

• Conversely, if we select $x_1 = b$, we can analyze how the first approximation leads to a subsequent approximation towards the actual root $w$. In this case, the function's characteristics near $b$ may lead to a more accurate second approximation.

Upon evaluating both scenarios, we find that selecting $x_1 = b$ is likely to yield a second approximation that will be closer to the solution $x = w$. Thus, the optimal initial approximation is:

Answer: $x_1 = b$ (Option B).