{Use of Tech} Modified Newton’s method The function f has a root of multiplicity 2 at r if f(r) = f'(r) = 0 and f"(r) ≠ 0. In this case, a slight modification of Newton’s method, known as the modified (or accelerated) Newton’s method, is given by the formula xₙ + 1 = xₙ - (2f(xₙ)/(f'(xₙ), for n = 0, 1, 2, . . . . This modified form generally increases the rate of convergence.
b. Apply Newton’s method and the modified Newton’s method using x₀ = 0.1 to find the value of x₃ in each case. Compare the accuracy of these values of x₃.

Verified step by step guidance

1. Define the function f(x) and its derivatives f'(x) and f''(x) that you will use for the calculations.
2. Calculate the first iteration using Newton's method: x₁ = x₀ - (f(x₀) / f'(x₀)) with x₀ = 0.1.
3. Continue to the second iteration for Newton's method: x₂ = x₁ - (f(x₁) / f'(x₁)).
4. Now, apply the modified Newton's method: calculate x₁ using the formula x₁ = x₀ - (2f(x₀) / f'(x₀)).
5. Proceed to the second iteration for the modified method: x₂ = x₁ - (2f(x₁) / f'(x₁)) and compare the results of x₃ from both methods.