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) = 1− 4/x², x ≠ 0

Verified step by step guidance

1. First, identify the critical points of the function by setting the derivative f'(x) = 1 - 4/x² equal to zero and solving for x. This will help us find where the function might have local maxima or minima.
2. Solve the equation 1 - 4/x² = 0 for x. This involves rearranging the equation to find x² = 4, and then taking the square root of both sides to find the possible values of x.
3. The solutions to x² = 4 are x = 2 and x = -2. These are the critical points where the function might have local maxima or minima.
4. To determine whether these points are local maxima or minima, use the second derivative test. Compute the second derivative f''(x) and evaluate it at x = 2 and x = -2.
5. If f''(x) > 0 at a critical point, the function has a local minimum there. If f''(x) < 0, the function has a local maximum. If f''(x) = 0, the test is inconclusive, and further analysis is needed.

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. In the given problem, setting f′(x) = 1 - 4/x² equal to zero helps identify critical points, which are essential for determining where the function might assume local extremum values.

Second Derivative Test

The second derivative test helps determine the nature of critical points. If the second derivative at a critical point is positive, the function has a local minimum there; if negative, a local maximum. Calculating the second derivative of f′(x) = 1 - 4/x² and evaluating it at critical points provides insight into whether these points are maxima or minima.

Behavior at Infinity

Understanding the behavior of a function as x approaches infinity or negative infinity can reveal asymptotic tendencies and help identify global behavior. For f′(x) = 1 - 4/x², analyzing the limit as x approaches infinity or zero can provide additional context about the function's overall behavior, complementing the analysis of local extrema.