4 A curve has equation \(y = 2 \ln ( k - 3 x ) + x ^ { 2 } - 3 x\), where \(k\) is a positive constant.
- Given that the curve has a point of inflection where \(x = 1\), show that \(k = 6\).
It is also given that the curve intersects the \(x\)-axis at exactly one point.
- Show by calculation that the \(x\)-coordinate of this point lies between 0.5 and 1.5 .
- Use the Newton-Raphson method, with initial value \(x _ { 0 } = 1\), to find the \(x\)-coordinate of the point where the curve intersects the \(x\)-axis, giving your answer correct to 5 decimal places. Show the result of each iteration to 6 decimal places.
- By choosing suitable bounds, verify that your answer to part (c) is correct to 5 decimal places.