Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:


a. What are the critical points of f?


f′(x) = x(x − 1)

Verified step by step guidance

1. To find the critical points of a function, we need to determine where its derivative is equal to zero or undefined. In this case, the derivative is given as f'(x) = x(x - 1).
2. Set the derivative equal to zero to find the critical points: x(x - 1) = 0.
3. Solve the equation x(x - 1) = 0 by setting each factor equal to zero: x = 0 and x - 1 = 0.
4. Solving these equations gives the critical points: x = 0 and x = 1.
5. Since the derivative is a polynomial, it is defined for all real numbers, so there are no points where the derivative is undefined. Therefore, the critical points are x = 0 and x = 1.

Key Concepts

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are significant because they can indicate local maxima, minima, or points of inflection. To find critical points, set the derivative equal to zero and solve for the variable, or identify where the derivative does not exist.

Derivative

The derivative of a function represents the rate of change or the slope of the function at any given point. It is a fundamental tool in calculus used to find critical points, analyze function behavior, and solve optimization problems. In this context, f′(x) = x(x − 1) is the derivative of the function f(x).

Factoring

Factoring is a mathematical process used to simplify expressions or solve equations by expressing a polynomial as a product of its factors. In the context of finding critical points, factoring the derivative, such as f′(x) = x(x − 1), helps identify the values of x that make the derivative zero, which are potential critical points.