In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
39. y = 8 / (x² + 4) (Witch of Agnesi)

Verified step by step guidance

1. Step 1: Analyze the function y = 8 / (x² + 4). Start by identifying its domain. Since the denominator x² + 4 is always positive and never zero, the function is defined for all real numbers x (domain: (-∞, ∞)).
2. Step 2: Find the first derivative y' to locate critical points and determine local extreme points. Use the quotient rule: if y = f(x)/g(x), then y' = (f'(x)g(x) - f(x)g'(x)) / [g(x)]². Here, f(x) = 8 and g(x) = x² + 4.
3. Step 3: Solve y' = 0 to find critical points. Set the numerator of the derivative equal to zero and solve for x. These x-values are potential locations for local maxima or minima. Verify these points by checking the sign of y' on intervals around each critical point.
4. Step 4: Find the second derivative y'' to determine concavity and locate inflection points. Use the quotient rule again to differentiate y'. Solve y'' = 0 to find potential inflection points, and verify by checking the sign of y'' on intervals around these points.
5. Step 5: Evaluate the function y at the critical points, inflection points, and endpoints (if applicable) to determine the coordinates of any absolute extreme points. Use this information to sketch the graph, noting the behavior of the function as x approaches ±∞ and the symmetry of the function (it is even, as y(x) = y(-x)).

Key Concepts

Graphing Rational Functions

Graphing rational functions involves understanding their behavior based on the numerator and denominator. Key steps include identifying asymptotes, intercepts, and the overall shape of the graph. For the function y = 8 / (x² + 4), recognizing that the denominator never equals zero helps in determining the function's continuity and limits.

Local and Absolute Extrema

Local extrema refer to points where a function reaches a maximum or minimum value within a specific interval, while absolute extrema are the highest or lowest points over the entire domain. To find these points, one typically uses the first derivative test to identify critical points and the second derivative test to determine their nature.

Inflection Points

Inflection points occur where the concavity of a function changes, which can be identified by analyzing the second derivative. For the function y = 8 / (x² + 4), finding inflection points involves setting the second derivative equal to zero and solving for x, indicating where the graph transitions from concave up to concave down or vice versa.