Math Problem Statement
Solution
Solution:
The problem involves analyzing the graph of the derivative of a function . We are asked to determine at which points the original function has a relative maximum.
Key Observations:
-
Behavior of the derivative :
- A relative maximum occurs at a point if:
- (i.e., a horizontal tangent in ).
- changes from positive to negative at .
This means we are looking for where the derivative crosses the x-axis (or touches it with a horizontal tangent) and the slope transitions from positive to negative.
- A relative maximum occurs at a point if:
Step-by-Step Analysis:
Looking at the graph of :
-
At :
- (it touches the x-axis).
- Sign change: changes from positive to negative.
- Therefore, has a relative maximum at .
-
At :
- (horizontal tangent line).
- Sign change: changes from positive to negative.
- Therefore, has a relative maximum at .
-
At :
- (horizontal tangent line).
- Sign change: changes from positive to negative.
- Therefore, has a relative maximum at .
Answer:
Since has relative maxima at , , and , the correct answer is:
(E) , 1, and 3
Do you need further clarifications or details on this?
Here are five related questions you might find helpful:
- How do you determine whether a critical point is a maximum or minimum using the first derivative test?
- What does it mean when the derivative touches but does not cross the x-axis?
- Can a function have a maximum or minimum at a point where is undefined?
- What is the significance of the second derivative in determining maxima and minima?
- How does the behavior of the derivative affect the concavity of the original function?
Tip: Always check where the derivative changes signs to determine relative maxima and minima—it provides a clearer picture than relying on values alone!
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivative
Relative Maximum
First Derivative Test
Formulas
f'(x) = 0 at critical points
f'(x) changes from positive to negative at a relative maximum
Theorems
First Derivative Test
Suitable Grade Level
Grades 11-12