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 − 1)(x + 2)(x − 3)

Verified step by step guidance

1. To find the critical points of the function f, we need to determine where the derivative f'(x) is equal to zero or undefined. Since f'(x) = (x - 1)(x + 2)(x - 3), we focus on where this expression equals zero.
2. Set f'(x) = 0: (x - 1)(x + 2)(x - 3) = 0. This equation is satisfied when any of the factors is zero.
3. Solve each factor for x: x - 1 = 0, x + 2 = 0, and x - 3 = 0. This gives us the potential critical points.
4. The solutions to these equations are x = 1, x = -2, and x = 3. These are the critical points of the function f.
5. Verify that these points are indeed critical points by ensuring that f'(x) changes sign around these values, indicating a change in the behavior of the function f.

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, which in this case involves solving f′(x) = (x − 1)(x + 2)(x − 3) = 0.

Factoring Polynomials

Factoring polynomials is a method used to simplify expressions and solve equations. It involves expressing a polynomial as a product of its factors. For the derivative f′(x) = (x − 1)(x + 2)(x − 3), the factors are already given, making it straightforward to find the roots by setting each factor equal to zero: x = 1, x = -2, and x = 3.

Roots of Equations

The roots of an equation are the values of the variable that satisfy the equation, making it equal to zero. In the context of derivatives, finding the roots of f′(x) helps identify the critical points of the function f(x). For f′(x) = (x − 1)(x + 2)(x − 3), the roots are x = 1, x = -2, and x = 3, which are the critical points of the function.