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.
53. y = x * √(8 - x²)

Verified step by step guidance

1. First, identify the domain of the function y = x * √(8 - x²). The expression under the square root, 8 - x², must be non-negative. Solve the inequality 8 - x² ≥ 0 to find the domain.
2. Next, find the first derivative of the function to identify critical points. Use the product rule and chain rule to differentiate y = x * √(8 - x²). Set the derivative equal to zero to find critical points, which may correspond to local extreme points.
3. Determine the second derivative to find inflection points. Differentiate the first derivative to obtain the second derivative. Set the second derivative equal to zero and solve for x to find potential inflection points.
4. Evaluate the function at the critical points and endpoints of the domain to find local extreme points. Compare these values to determine if they are local maxima or minima.
5. Finally, evaluate the function at the critical points and endpoints of the domain to find absolute extreme points. Compare these values to determine the absolute maximum and minimum values of the function within the domain.

Key Concepts

Graphing Functions

Graphing functions involves plotting the curve of a function on a coordinate plane to visualize its behavior. This includes identifying key features such as intercepts, asymptotes, and the general shape of the graph. Understanding how to graph functions is essential for analyzing their properties and behavior over different intervals.

Local Extreme Points

Local extreme points are points on the graph where the function reaches a local maximum or minimum. These occur where the derivative of the function equals zero or is undefined, indicating a change in direction. Identifying these points helps in understanding the function's behavior in specific regions and is crucial for optimization problems.

Inflection Points

Inflection points are points on the graph where the curvature changes direction, from concave up to concave down or vice versa. These are found by analyzing the second derivative of the function. Inflection points provide insight into the function's concavity and are important for understanding the overall shape and behavior of the graph.