The graph of the derivative, f '(x), is given. Determine the x-coordinates of all points of inflection of f(x), if any. (Assume that f(x) is defined and continuous everywhere in [-3, 3]. If there are more answer blanks than inflection points, enter NONE in any remaining spaces)
(x1) = Incorrect: Your answer is incorrect. (smaller value)
(x2) = Correct: Your answer is correct. (larger value)
Look at the graph. f(x) has points of inflection where f''(x) = 0.
But f''(x)=0 means that f'(x) has a min or max. So, observe the x-values where the given graph has a minimum or maximum. That is where f(x) has a point of inflection.
To determine the x-coordinates of points of inflection of f(x) using the graph of the derivative, we need to look for the places on the graph of f '(x) where it changes sign from positive to negative or negative to positive.
Here's how you can find the x-coordinates of the points of inflection:
1. Locate the x1 value on the graph of f '(x) where the derivative changes sign from positive to negative. This change in sign indicates a possible point of inflection for f(x).
2. To determine the x1-coordinate, we need to find the corresponding x-value on the x-axis of the graph of f '(x). Look for the x-value where the graph of f '(x) intersects the x-axis.
3. Similarly, locate the x2 value on the graph of f '(x) where the derivative changes sign from negative to positive. This change in sign also indicates a possible point of inflection for f(x).
4. To determine the x2-coordinate, find the corresponding x-value on the x-axis where the graph of f '(x) intersects the x-axis for the second time.
5. Enter the x1 value as the "smaller value" and the x2 value as the "larger value" in the given answer blanks.
If there are no points of inflection for f(x), enter NONE in both answer blanks.
Remember to consider the given interval [-3, 3] when determining the points of inflection.