Question

Analyzing Functions from DerivativesAnswer 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)

Accepted Answer

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. 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. 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. 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. 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.

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)

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Concetti chiave

Critical Points

First Derivative Test

Sign Analysis of Derivative