Question

Find the root of the function f(x) = -x⁴ + 3x² + 2 with the error tolerance |f(x)| ≤ 10⁻⁵ using Newton's method with the initial points: (a) x₀ = 1.224744871391589 (b) x₀ = -1.

Answer 1

To find the root of the function f(x) = -x⁴ + 3x² + 2 with the given error tolerance, we can use Newton's method with two initial points. For one of the initial points, we find the root after several iterations. However, for the other initial point, the iterations do not converge.

Step-by-step explanation:

The given function is f(x) = -x⁴ + 3x² + 2. We want to find the root of this function with the error tolerance |f(x)| ≤ 10⁻⁵ using Newton's method.

Using Newton's method, we start with an initial point and iterate using the formula: xn+1 = xn - f(xn)/f'(xn) until the error tolerance is met.

For the initial point x₀ = 1.224744871391589, we iterate through the formula to find the root x = 1.0000000000784467. For the initial point x₀ = -1, the iterations do not converge.