{Use of Tech} Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = x³ - 3x² + x + 1

Verified step by step guidance

1. Start by rewriting the equation for fixed points, which is f(x) = x. For the given function, set x³ - 3x² + x + 1 = x.
2. Simplify the equation to find the roots: x³ - 3x² + x + 1 - x = 0, which simplifies to x³ - 3x² + 1 = 0.
3. Next, perform a preliminary analysis by evaluating the function at several integer values to identify potential fixed points. Calculate f(-1), f(0), f(1), f(2), and f(3).
4. Graph the function f(x) = x³ - 3x² + 1 and the line y = x on the same set of axes to visually identify where the two graphs intersect, indicating the fixed points.
5. Use numerical methods such as the Newton-Raphson method or bisection method to refine the approximations of the fixed points found from the graph.