if ƒ(x) = 1 / (3x⁴ + 5) , it can be shown that ƒ'(x) = 12x³ / (3x⁴ + 5)² and ƒ"(x) = 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ . Use these functions to complete the following steps.


d. Identify the local extreme values and inflection points of ƒ .

Verified step by step guidance

1. To find the local extreme values, we need to analyze the first derivative, ƒ'(x). Set ƒ'(x) = 0 to find critical points. This involves solving the equation 12x³ / (3x⁴ + 5)² = 0, which simplifies to finding when the numerator, 12x³, is zero.
2. The critical points occur when 12x³ = 0, which simplifies to x = 0. This is the only critical point since the denominator (3x⁴ + 5)² is never zero.
3. To determine if x = 0 is a local maximum or minimum, use the second derivative test. Evaluate ƒ''(x) at x = 0. If ƒ''(0) > 0, x = 0 is a local minimum; if ƒ''(0) < 0, x = 0 is a local maximum.
4. Next, identify the inflection points by setting the second derivative, ƒ''(x), equal to zero. Solve 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ = 0. This requires finding when the numerator, 180x² (x² + 1) (x + 1) (x - 1), is zero.
5. The potential inflection points occur when x² = 0, x² + 1 = 0, x + 1 = 0, or x - 1 = 0. Solve these equations to find x = 0, x = -1, and x = 1. Check the sign change of ƒ''(x) around these points to confirm they are inflection points.

Key Concepts

Local Extreme Values

Local extreme values refer to the points in a function where it reaches a local maximum or minimum. These points occur where the first derivative, ƒ'(x), is equal to zero or undefined. To identify these points, one typically analyzes the sign changes of the first derivative around critical points, which helps determine whether the function is increasing or decreasing.

Inflection Points

Inflection points are points on the graph of a function where the concavity changes, which can be identified by examining the second derivative, ƒ''(x). Specifically, an inflection point occurs where ƒ''(x) equals zero or is undefined, and the sign of ƒ''(x) changes around that point. This indicates a transition in the curvature of the graph, which is crucial for understanding the function's behavior.

First and Second Derivatives

The first derivative of a function, ƒ'(x), provides information about the function's rate of change and is essential for finding local extrema. The second derivative, ƒ''(x), offers insights into the function's concavity and helps identify inflection points. Together, these derivatives are fundamental tools in calculus for analyzing the behavior of functions and understanding their graphical representations.