Introduction
This topic focuses on the logical connections between , , and . The essential knowledge tells us that key features of the graphs of , , and are related to one another. Understanding these relationships allows us to make justified conclusions about one function when given information about another.
Key Relationships Summary
The Connection Table
Reasoning from f to f' and f''
Given Properties of f
Example 1:
If has a local minimum at , what can be concluded about and at ?
About f':
- (horizontal tangent at the minimum)
- changes from negative to positive at
About f'':
- (concave up at a minimum)
Alternatively, using the First Derivative Test, changes from to , so is increasing at , which means .
Inflection Point Connections
Example 2:
If has an inflection point at , what must be true about and ?
About f'':
- must change sign at
- Typically (though it could be undefined)
About f':
- has a local maximum or minimum at
- This is because changes from increasing to decreasing (or vice versa) when changes sign
Reasoning from f' to f and f''
Given Properties of f'
Example 3:
If for all in and is decreasing on this interval, what can be concluded about ?
About f:
- is increasing on because
- is concave down on because is decreasing
About f'':
- on because is decreasing
Example 4:
If has a local maximum at , what occurs on the graph of at ?
About f:
- has an inflection point at
- The concavity of changes at
Reasoning:
- has a local maximum means changes from increasing to decreasing
- This means changes from positive to negative
- A sign change in indicates an inflection point on
Reasoning from f'' to f and f'
Given Properties of f''
Example 5:
If for all in an interval, what can be concluded about and ?
About f:
- is concave down on the interval
About f':
- is decreasing on the interval
- The slopes of tangent lines to are getting less steep (the absolute value of derivative is becoming smaller)
Sign Changes in f''
Example 6:
If , where do inflection points occur on the graph of ?
Analyze sign changes:
- For : (concave up)
- For : (concave down)
- For : (concave up)
Conclusion: Inflection points at and (where changes sign).
At these points, has local extrema:
- has a local maximum at (where changes from to )
- has a local minimum at (where changes from to )
Justification and Reasoning
Complete Justifications
On the AP exam, you must justify conclusions using proper mathematical reasoning.
Example 7:
Given that and , justify that has a local maximum at .
Justification:
Since , is a critical point of .
Since , the function is concave down at .
By the Second Derivative Test, has a local maximum at .
Alternative justification using First Derivative Test:
Since , is decreasing at .
Because and is decreasing, changes from positive to negative at .
Therefore, by the First Derivative Test, has a local maximum at .