Given the graph of f'(x) with f(x) continuous on all reals, find the intervals f(x) is concave up and down and find the x values of the points of inflection.
concave up means f" > 0
That means f' is increasing.
inflection points are where f"=0 and f'≠0
You need to make sure that there are actually inflection points by plugging in values near the x solution into the function of x. Inflection points may or may not be present despite finding x solutions or undefined "points" when the second derivative is set to zero.
To find the intervals where a function is concave up or down and the x-values of the inflection points, we need to analyze the graph of the second derivative, f''(x).
1. First, identify the critical points of f''(x) by finding where f''(x) = 0 or does not exist. These points might correspond to inflection points.
2. Next, examine the behavior of f''(x) around the critical points by considering the sign of f''(x) in different intervals.
- If f''(x) > 0 in an interval, then f(x) is concave up in that interval.
- If f''(x) < 0 in an interval, then f(x) is concave down in that interval.
3. Finally, use the sign changes in f''(x) to determine the locations of inflection points. An inflection point occurs where the concavity changes, which means f''(x) changes sign.
Now let's interpret this information based on the given information that f'(x) is continuous on all reals and the graph of f'(x) is provided.
1. Locate the critical points of f''(x) by finding where f''(x) = 0 or does not exist.
- If f'(x) is increasing on an interval, then f''(x) is positive in that interval.
- If f'(x) is decreasing on an interval, then f''(x) is negative in that interval.
- The points where f'(x) has vertical tangents or sharp corners correspond to where f''(x) does not exist.
2. Once you have identified the critical points, analyze the sign of f''(x) in each interval to determine concavity:
- If f''(x) > 0 in an interval, then f(x) is concave up in that interval.
- If f''(x) < 0 in an interval, then f(x) is concave down in that interval.
3. Inflection points occur where f''(x) changes sign. Look for sign changes in f''(x) to identify these points.
By following these steps and analyzing the given graph of f'(x), you should be able to determine the intervals where f(x) is concave up and down and find the x-values of the points of inflection.