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.
56. y = x² + 2/x

Verified step by step guidance

1. First, identify the domain of the function y = x² + 2/x. Since the term 2/x is undefined for x = 0, the domain is all real numbers except x = 0.
2. Next, find the first derivative y' to determine critical points and local extrema. Use the quotient rule for differentiation: y' = d/dx (x² + 2/x).
3. Set the first derivative y' equal to zero to find critical points. Solve the equation y' = 0 for x to find potential local extrema.
4. Find the second derivative y'' to determine concavity and inflection points. Use the derivative of y' to find y'' and set y'' = 0 to find potential inflection points.
5. Evaluate the function y at the critical points and endpoints (if any) to find absolute extrema. Compare these values to determine the absolute maximum and minimum values of the function on its domain.

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, minima, or inflection points. To find them, compute the derivative of the function and solve for the values of x where the derivative equals zero or does not exist.

Second Derivative Test

The second derivative test helps determine the concavity of a function and identify inflection points. If the second derivative is positive at a critical point, the function is concave up, indicating a local minimum. If negative, the function is concave down, indicating a local maximum. A change in sign of the second derivative indicates an inflection point.

Absolute Extrema

Absolute extrema are the highest or lowest points over the entire domain of a function. To find them, evaluate the function at critical points and endpoints of the domain. Compare these values to determine the absolute maximum and minimum. This is crucial for understanding the overall behavior of the function across its domain.