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In the Newton-Raphson method, an initial guess of \(x_0 = 2\) is made and the sequence \(x_0, x_1, x_2, \ldots\) is obtained for the equation \(0.75x^3 - 2x^2 - 2x + 4 = 0\). Consider the statements: (I) \(x_3 = 0\). (II) The method converges to a solution in a finite number of iterations. Which of the following is true?

1. Only I
2. Only II
3. Both I and II
4. Neither I nor II

Correct answer: Both I and II

Solution

Applying Newton-Raphson to the given polynomial from x0 = 2 leads to successive iterates that reach a root exactly in a finite number of steps. In particular, the third iterate becomes 0, so statement I is true. Since a root is obtained exactly, the method converges in a finite number of iterations, so statement II is also true.

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