In the Newton-Raphson method, an initial guess of $x_0 = 2$ is made and the sequence $x_0, x_1, x_2 \dots$ is obtained for the function $$0.75x^3 - 2x^2 - 2x + 4 = 0$$ 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?

Let $f(x) = 0.75x^3 - 2x^2 - 2x + 4 = \frac{3}{4}x^3 - 2x^2 - 2x + 4$.

The derivative is:

The Newton-Raphson iteration formula is:

Iteration 1: For $x_0 = 2$:

Iteration 2: For $x_1 = 0$:

Iteration 3: Since $x_2 = x_0 = 2$, we get:

Thus, statement (I) is TRUE as $x_3 = 0$.

Statement (II) is FALSE because the sequence oscillates endlessly between $2$ and $0$ ($2, 0, 2, 0, \dots$) and never converges to a root.

Therefore, only statement I is true.