{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) = cos x

Verified step by step guidance

1. Understand the definition of a fixed point: A fixed point of a function f(x) is a value x such that f(x) = x. For the function f(x) = \cos(x), we need to find x such that \cos(x) = x.
2. Perform a preliminary analysis: Consider the range of the cosine function, which is [-1, 1]. This means any fixed point must lie within this interval.
3. Graph the functions y = \cos(x) and y = x: By graphing these two functions, you can visually identify where they intersect. These intersection points are the fixed points.
4. Use numerical methods for better approximation: Since \cos(x) is a transcendental function, finding an exact solution algebraically might be challenging. Use numerical methods like the Newton-Raphson method or fixed-point iteration to approximate the solution.
5. Verify the solution: Once you have an approximate solution, substitute it back into the equation \cos(x) = x to check if it holds true within an acceptable margin of error.