Math Problem Statement
Solve and explain and sketch f(x)
Solution
We are given the graph of , the derivative of , and we need to determine:
- Where is increasing.
- Where has an inflection point.
Understanding the graph:
- is positive where the graph is above the x-axis, which means is increasing in those intervals.
- is negative where the graph is below the x-axis, which means is decreasing in those intervals.
- Inflection points occur where changes sign, which indicates a change in the concavity of .
First: The function is increasing on...
The function is increasing where , meaning the curve of is above the x-axis.
From the graph:
- on two intervals:
- (between 2 and 3)
- (from 4 onwards)
Thus, the correct answer is:
- is increasing on both intervals and . So, the answer is (d) both (a) and (b).
Second: The function has an inflection point at...
An inflection point occurs where changes sign. This happens at points where crosses the x-axis.
From the graph:
- changes sign at , , and .
Thus, the function has inflection points at , , and . So, the correct answer is (d) All the previous.
Sketching
To sketch , we can infer the following from :
- is increasing on and .
- is decreasing on and .
- has inflection points at , , and .
Based on this, the graph of will have:
- A decreasing slope from ,
- A local minimum at ,
- Increasing slope between ,
- A local maximum at ,
- Decreasing slope between ,
- A local minimum at ,
- Increasing slope for .
Do you want me to generate a sketch of ?
Follow-up questions:
- What does it mean if the derivative crosses the x-axis?
- How does the sign of affect the behavior of ?
- What would happen if does not change sign at all?
- How can we find concavity from the second derivative?
- What are the implications of an inflection point on the graph of ?
Tip:
When analyzing the graph of , always focus on where it crosses the x-axis and its sign to understand the behavior of .
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Inflection Points
Formulas
f'(x) > 0 implies f(x) is increasing
f'(x) = 0 or changes sign implies a critical point or inflection point
Theorems
First Derivative Test
Inflection Point Condition
Suitable Grade Level
Grades 10-12
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