Question

Find the intervals on which the graph of 'f' is concave upward, the intervals on which the graph of 'f' is concave downward, and the 'x, y coordinates of the inflection.

Answer 1

To determine the concavity of the graph of 'f' and its inflection points:

• Concave upward intervals: [Interval 1], [Interval 2]
• Concave downward intervals: [Interval 3]
• Inflection point coordinates: (x-coordinate, y-coordinate)

Step-by-step explanation:

The concavity of a function 'f' changes where its second derivative switches signs. To find these intervals, compute the second derivative of 'f' and then identify where it equals zero or is undefined.

1. Finding the Second Derivative: Differentiate 'f' with respect to 'x' to obtain the first derivative, then differentiate the first derivative to acquire the second derivative.
2. Identifying Intervals: Examine where the second derivative is positive to identify concave upward intervals and where it is negative for concave downward intervals. These intervals correspond to where the concavity changes.
3. Locating Inflection Points: Locate points where the second derivative equals zero or is undefined. Evaluate these x-values in the original function 'f' to obtain their respective y-values, determining the coordinates of the inflection points.

Once the intervals of concavity are identified, and inflection points are located, you'll have a comprehensive understanding of the function's behavior regarding concavity and inflection points. Remember to verify the nature of the inflection points by analyzing the behavior of the function on both sides of these points. This analysis helps in accurately determining whether the function changes concavity at those specific x-values.