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) = 1− 4/x², x ≠ 0

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Identify the critical points by setting the derivative \( f'(x) = 1 - \frac{4}{x^2} \) equal to zero and solving for \( x \). This will help determine where the function changes from increasing to decreasing or vice versa.

Solve the equation \( 1 - \frac{4}{x^2} = 0 \) to find the critical points. Rearrange the equation to \( \frac{4}{x^2} = 1 \) and solve for \( x \).

Determine the sign of \( f'(x) \) on the intervals defined by the critical points. Choose test points in each interval and substitute them into \( f'(x) \) to see if the derivative is positive or negative.

If \( f'(x) > 0 \) on an interval, then \( f(x) \) is increasing on that interval. If \( f'(x) < 0 \), then \( f(x) \) is decreasing on that interval.

Summarize the intervals where \( f(x) \) is increasing and decreasing based on the sign of \( f'(x) \) in each interval.

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Derivative and Function Behavior

The derivative of a function, f'(x), provides information about the function's rate of change. If f'(x) > 0 on an interval, the function is increasing there; if f'(x) < 0, the function is decreasing. Understanding the sign of the derivative is crucial for determining where the function is increasing or decreasing.

Critical Points and Intervals

Critical points occur where the derivative is zero or undefined, indicating potential changes in the function's behavior. To find intervals of increase or decrease, identify these points and test the sign of the derivative in the intervals they create. This helps in understanding the function's behavior across its domain.

Open Intervals

Open intervals are ranges of x-values where the function's behavior is consistent, either increasing or decreasing. They do not include their endpoints, which is important when analyzing functions, especially when the derivative is undefined at certain points, like x = 0 in this case.

Analyzing Functions from DerivativesAnswer 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) = 1− 4/x², x ≠ 0