{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51).​​ 


d. Find the number and location of the fixed points of g for a = 2, 3, and 4 on the interval 0 ≤ x ≤ 1. 

Verified step by step guidance

1. Understand the concept of a fixed point: A fixed point of a function f(x) is a value x such that f(x) = x. This means that when you substitute x into the function, the output is the same as the input.
2. Set up the equation for fixed points: For the function f(x) = ax(1 - x), we need to solve the equation ax(1 - x) = x to find the fixed points.
3. Rearrange the equation: Start by expanding the left side to get ax - ax^2 = x. Then, move all terms to one side to form a quadratic equation: ax - ax^2 - x = 0.
4. Factor the quadratic equation: Factor the equation ax - ax^2 - x = 0 to find the values of x. This can be done by factoring out x, giving x(a - ax - 1) = 0.
5. Solve for x: The factored equation x(a - ax - 1) = 0 gives two potential solutions: x = 0 and a - ax - 1 = 0. Solve the second equation for x to find the other fixed points, and analyze these solutions for different values of a (2, 3, and 4) within the interval 0 ≤ x ≤ 1.

Key Concepts

Fixed Points

A fixed point of a function f(x) is a value x such that f(x) = x. This means that when the function is applied to this value, it returns the same value. Finding fixed points often involves solving the equation f(x) - x = 0. In the context of the given function, identifying fixed points helps understand the behavior of the function within a specified interval.

Quadratic and Quartic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax^2 + bx + c. Quartic functions are of degree four, represented as f(x) = ax^4 + bx^3 + cx^2 + dx + e. The behavior of these functions, including their fixed points, can be analyzed using their graphs, derivatives, and algebraic properties, which are essential for solving the problem at hand.

Interval Analysis

Interval analysis involves examining the behavior of functions within a specific range, in this case, 0 ≤ x ≤ 1. This is crucial for determining the existence and location of fixed points, as the function's behavior may vary significantly outside this interval. By restricting the analysis to a defined interval, one can apply techniques such as the Intermediate Value Theorem to ascertain the number of fixed points.