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)²(x + 2)

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 zero or undefined. Start by setting f'(x) = (x - 1)²(x + 2) equal to zero.
2. Solve the equation (x - 1)²(x + 2) = 0. This equation is satisfied when either (x - 1)² = 0 or (x + 2) = 0. Solve these separately to find the critical points.
3. For (x - 1)² = 0, solve for x to get x = 1. For (x + 2) = 0, solve for x to get x = -2. Thus, the critical points are x = 1 and x = -2.
4. To determine whether these critical points are local maxima, minima, or neither, use the first derivative test. Evaluate the sign of f'(x) around the critical points. Choose test points in the intervals created by the critical points: (-∞, -2), (-2, 1), and (1, ∞).
5. For each interval, pick a test point and substitute it into f'(x) to determine the sign. If f'(x) changes from positive to negative at a critical point, it is a local maximum. If it changes from negative to positive, it is a local minimum. If there is no sign change, it 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)²(x + 2) = 0, leading to critical points at x = 1 and x = -2.

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, one can infer the behavior of the function. If f′ changes from positive to negative at a point, it's a local maximum; if it changes from negative to positive, it's a local minimum.

Behavior of Polynomial Functions

Understanding the behavior of polynomial functions is crucial for analyzing their derivatives. The degree and leading coefficient of a polynomial influence its end behavior and the number of turning points. For f′(x) = (x − 1)²(x + 2), the derivative is a cubic polynomial, indicating up to two turning points, which correspond to potential local extrema.