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

a. Verify that 0 is a root of multiplicity 2 of the function f(x) = e^{2 sin x} - 2x - 1.

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 each value of x₃. 

c. Consider the function f(x) = 8x²/3x² + 1 given in Example 4. Use the modified Newton’s method to find the value of x₃ using x₀ = 0.15. Compare this value to the value of x3 found in Example 4 with x₀ = 0.15.

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2 f(x„) f'(x„)° for n = 0, 1, 2, . . .. Xn+1 = Xn