Math Problem Statement
find the intervals of concavity y=sin(2x) [-pi,pi]
Solution
To determine the intervals of concavity for the function over the interval , we need to find the second derivative of the function and analyze its sign.
Step 1: Find the first and second derivatives
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First derivative :
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Second derivative :
Step 2: Determine where the second derivative changes sign
The concavity of the function is determined by the sign of the second derivative:
- If , the function is concave up.
- If , the function is concave down.
Since , we analyze where is positive or negative.
- when (concave up).
- when (concave down).
Step 3: Solve for critical points where
The solutions to are: On the interval , the critical points are:
Step 4: Analyze the sign of on intervals
Now, we check the sign of on the intervals determined by the critical points :
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Interval : For , , and is negative. Therefore, , so the function is concave up on this interval.
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Interval : For , , and is positive. Therefore, , so the function is concave down on this interval.
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Interval : For , , and is positive. Therefore, , so the function is concave down on this interval.
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Interval : For , , and is negative. Therefore, , so the function is concave up on this interval.
Step 5: Conclusion
The concavity of on is:
- Concave up on and .
- Concave down on and .
Summary of concavity intervals:
- Concave up:
- Concave down:
Would you like more details on the steps or a different aspect of this problem? Feel free to ask!
Here are some related questions to explore:
- How do the concavity intervals change for a function with a higher-degree sine or cosine term?
- What is the relationship between the second derivative and inflection points?
- How do you find the concavity of trigonometric functions involving other angles, such as ?
- How does the graph of look in terms of concavity?
- Can the second derivative help in identifying local maxima and minima in the function?
Tip: When analyzing concavity, always check the sign of the second derivative at the critical points, and test within the intervals defined by these points to determine the concavity behavior!
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Math Problem Analysis
Mathematical Concepts
Calculus
Concavity
Derivatives
Trigonometric Functions
Formulas
First derivative: y' = d/dx(sin(2x)) = 2cos(2x)
Second derivative: y'' = d/dx(2cos(2x)) = -4sin(2x)
Theorems
Second Derivative Test
Concavity and the Second Derivative
Suitable Grade Level
Grades 11-12
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