Question

In the Newton-Raphson method, an initial guess of x₀ = 2 is made and the sequence x₀, x₁, x₂ …

is obtained for the function

0.75x³ – 2x² – 2x + 4 = 0

Consider the statements

(I) x₃ = 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

Answer 1

Correct Answer - Option 1 : Only I

In Newton-Raphson method

\({x_{n + 1}} = {x_n} - \frac{{f\left( {{x_n}} \right)}}{{f'\left( {{x_n}} \right)\;}}\;\)

Calculation:

f(x) = 0.75x³ – 2x² – 2x + 4 = 0

f'(x) = 2.25×x² – 4x - 2

x₁ = 2 - (0.75× 2³ – 2×2² – 2× 2 + 4) ÷ (2.25 × 2² – 4× 2 - 2)

  = 2 - (-2/-1)

  = 0

x₂ = 0 - (4 ÷ -2) = 2

x₃ = 0

x = 0 and x = 2 repeatedly and it never converges.