{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = x⁵/5 - x³/4 - 1/20

Verified step by step guidance

1. Step 1: Begin by performing a preliminary analysis of the function f(x) = \frac{x^5}{5} - \frac{x^3}{4} - \frac{1}{20}. Identify the behavior of the function by considering its end behavior and any obvious roots or symmetry. This can be done by analyzing the leading term \frac{x^5}{5}, which suggests that as x approaches positive or negative infinity, f(x) will also approach positive or negative infinity respectively.
2. Step 2: Graph the function f(x) to visually identify potential roots. Look for points where the graph crosses the x-axis, as these are the x-values where f(x) = 0. Use this graph to make educated guesses for initial approximations of the roots.
3. Step 3: Apply Newton's method to refine these initial approximations. Newton's method uses the formula x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}, where f'(x) is the derivative of f(x). Calculate the derivative f'(x) = x^4 - \frac{3x^2}{4}.
4. Step 4: Choose an initial approximation x_0 based on the graph and apply the Newton's method formula iteratively. For each iteration, compute x_{n+1} using the current approximation x_n, the function value f(x_n), and the derivative f'(x_n). Repeat this process until the values converge to a stable root.
5. Step 5: Verify the roots found by substituting them back into the original function f(x) to ensure that they satisfy f(x) = 0. If necessary, adjust the initial approximations and repeat the process to find additional roots, ensuring all roots are identified.