Final answer:
To find the intervals of increasing and decreasing, the asymptotes, the local maximum and minimum values, the intervals of concavity, and the inflection points of the function f(x) = x² / (x² + 3), we need to analyze its derivative and second derivative. Then, we can sketch the graph using the gathered information.
Step-by-step explanation:
a) Intervals of Increasing and Decreasing:
To find the intervals on which f is increasing and decreasing, we need to find the critical points of the function. The critical points occur where the derivative is zero or undefined. To find the critical points, we can find the derivative of f and set it equal to zero.
b) Asymptotes:
To find the asymptotes, we need to consider the behavior of the function as x approaches positive or negative infinity.
c) Local Maximum and Minimum Values:
To find the local maximum and minimum values, we need to examine the critical points and the behavior of the function in the intervals determined by the critical points.
d) Intervals of Concavity and Inflection Points:
To find the intervals of concavity and the inflection points, we need to analyze the second derivative of f. The concavity of the function changes at points where the second derivative changes sign, indicating the presence of inflection points.
d) Sketching the Graph:
To sketch the graph of f, we can use the information gathered from the previous steps to plot the critical points, the behavior of the function, the asymptotes, and the concavity.