Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
c. At what points, if any, does f assume local maximum or minimum values?


f′(x) = (x − 1)(x + 2)(x − 3)

Verified step by step guidance

1. To find the points where the function f assumes local maximum or minimum values, we need to identify the critical points. Critical points occur where the derivative f'(x) is equal to zero or is undefined. In this case, f'(x) = (x - 1)(x + 2)(x - 3) is a polynomial, so it is defined everywhere. Set f'(x) = 0 to find the critical points.
2. Solve the equation (x - 1)(x + 2)(x - 3) = 0. This equation is satisfied when any of the factors is zero. Therefore, the critical points are x = 1, x = -2, and x = 3.
3. Next, determine whether each critical point is a local maximum, local minimum, or neither. This can be done using the First Derivative Test. Evaluate the sign of f'(x) around each critical point.
4. Choose test points in the intervals determined by the critical points: (-∞, -2), (-2, 1), (1, 3), and (3, ∞). For each interval, pick a test point and substitute it into f'(x) to determine the sign of the derivative in that interval.
5. Based on the sign changes of f'(x) around each critical point, conclude whether each point is a local maximum, local minimum, or neither. If f'(x) changes from positive to negative at a critical point, it is a local maximum. If f'(x) changes from negative to positive, it is a local minimum. If there is no sign change, the point is neither.

Key Concepts

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are potential locations for local maxima or minima. To find them, set the derivative equal to zero and solve for x. In this case, f′(x) = (x − 1)(x + 2)(x − 3) = 0, so the critical points are x = 1, x = -2, and x = 3.

First Derivative Test

The First Derivative Test helps determine whether a critical point is a local maximum, minimum, or neither. By analyzing the sign changes of the derivative around the critical points, we can infer the behavior of the function. If f′ changes from positive to negative at a critical point, it's a local maximum; if it changes from negative to positive, it's a local minimum.

Sign Analysis of Derivative

Sign analysis involves examining the intervals between critical points to determine the sign of the derivative. This helps in understanding the function's increasing or decreasing behavior. For f′(x) = (x − 1)(x + 2)(x − 3), evaluate the sign of f′ in intervals like (-∞, -2), (-2, 1), (1, 3), and (3, ∞) to apply the First Derivative Test effectively.