Math Problem Statement
Solution
To determine which of the statements are true based on the graph of :
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Relative Min at :
- A relative minimum of occurs where changes from negative to positive.
- At , is increasing (it changes from negative to positive).
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Point of Inflection at :
- A point of inflection of occurs where (the second derivative) changes sign, which corresponds to having a local extremum (either a max or min).
- At , has a local minimum, indicating that changes sign, so there is a point of inflection at .
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Concave Down for :
- is concave down where , which corresponds to being decreasing.
- For , is indeed decreasing, indicating is concave down in this interval.
Based on the above analysis, all three statements are true:
- I is true because has a relative minimum at .
- II is true because the graph of has a point of inflection at .
- III is true because the graph of is concave down for .
Therefore, the correct answer is:
All three statements (I, II, and III) are true.
Would you like more details or have any questions?
Here are some questions you might ask next:
- Can you explain how to identify points of inflection on a graph of ?
- How do you determine concavity from the graph of ?
- What are the key features to look for on the graph of a derivative?
- Can you give an example of how to find relative extrema using the first derivative?
- How does the behavior of relate to the concavity of ?
Tip: When analyzing the graph of a derivative, remember that points where the derivative changes sign can indicate relative extrema of the original function.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Inflection Points
Concavity
Formulas
First derivative test
Second derivative test
Theorems
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Suitable Grade Level
Advanced High School and College
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