Analyzing Functions from Derivatives

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

b. On what open intervals is f increasing or decreasing?

f′(x) = (x − 1)²(x + 2)

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First, identify the critical points of the function by setting the derivative f'(x) = (x - 1)²(x + 2) equal to zero and solving for x. This will help us find where the function changes from increasing to decreasing or vice versa.

The critical points occur where f'(x) = 0. Solve the equation (x - 1)²(x + 2) = 0. This gives us x = 1 and x = -2 as critical points.

Next, determine the sign of f'(x) in the intervals defined by the critical points. These intervals are (-∞, -2), (-2, 1), and (1, ∞). Choose test points from each interval to evaluate the sign of f'(x).

For the interval (-∞, -2), choose a test point like x = -3. Substitute x = -3 into f'(x) to determine if f'(x) is positive or negative, indicating whether f is increasing or decreasing.

Repeat the process for the intervals (-2, 1) and (1, ∞) using test points like x = 0 and x = 2, respectively. Evaluate the sign of f'(x) at these points to determine the behavior of f in each interval.

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Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are essential for determining where a function changes from increasing to decreasing or vice versa. For the derivative f′(x) = (x − 1)²(x + 2), the critical points are found by setting the derivative equal to zero, resulting in x = 1 and x = -2.

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Critical Points

Increasing and Decreasing Intervals

A function is increasing on intervals where its derivative is positive and decreasing where its derivative is negative. By analyzing the sign of f′(x) = (x − 1)²(x + 2) around the critical points, we can determine the intervals of increase and decrease. This involves testing values in the intervals defined by the critical points to see where the derivative changes sign.

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Determining Where a Function is Increasing & Decreasing

Sign Analysis

Sign analysis involves evaluating the sign of the derivative in different intervals to determine the behavior of the function. For f′(x) = (x − 1)²(x + 2), we test intervals around x = 1 and x = -2. Since (x − 1)² is always non-negative, the sign of f′(x) is determined by (x + 2), which changes sign at x = -2, indicating where the function increases or decreases.

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