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Μάθημα: Διαφορικός Λογισμός > Ενότητα 5
Μάθημα 7: Analyzing concavity and inflection points- Analyzing concavity (algebraic)
- Inflection points (algebraic)
- Mistakes when finding inflection points: second derivative undefined
- Mistakes when finding inflection points: not checking candidates
- Analyzing the second derivative to find inflection points
- Analyze concavity
- Find inflection points
- Concavity review
- Inflection points review
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Analyzing the second derivative to find inflection points
Learn how the second derivative of a function is used in order to find the function's inflection points. Learn which common mistakes to avoid in the process.
We can find the inflection points of a function by analyzing its second derivative.
Example: Finding the inflection points of
Step 1: Finding the second derivative
To find the inflection points of , we need to use :
Step 2: Finding all candidates
Similar to critical points, these are points where or where is undefined.
Step 3: Analyzing concavity
Interval | Test | Conclusion | |
---|---|---|---|
Step 4: Finding inflection points
Now that we know the intervals where is concave up or down, we can find its inflection points (i.e. where the concavity changes direction).
is concave down before , concave up after it, and is defined at . So has an inflection point at . is concave up before and after , so it doesn't have an inflection point there.
We can verify our result by looking at the graph of .
Common mistake: not checking the candidates
Remember: We must not assume that any point where (or where is undefined) is an inflection point. Instead, we should check our candidates to see if the second derivative changes signs at those points and the function is defined at those points.
Common mistake: not including points where the derivative is undefined
Remember: Our candidates for inflection points are points where the second derivative is equal to zero and points where the second derivative is undefined. Ignoring points where the second derivative is undefined will often result in a wrong answer.
Common mistake: looking at the first derivative instead of the second derivative
Remember: When looking for inflection points, we must always analyze where the second derivative changes its sign. Doing this for the first derivative will give us relative extremum points, not inflection points.
Want more practice? Try this exercise.
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