Math Problem Statement
Identify the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.) h(x) = cos(3x/2)
, 0 < x < 2𝜋
Solution
To determine the intervals where is increasing or decreasing, we need to analyze the behavior of its derivative within the given interval .
Step 1: Differentiate
The derivative of with respect to is:
Step 2: Set the Derivative Equal to Zero
To find the critical points where the function could change from increasing to decreasing or vice versa, we set : This simplifies to:
Within the interval , we find the values of for which : Solving for , we get: Now, find the values of within by substituting integer values for :
- :
- :
- :
Thus, the critical points within the interval are and .
Step 3: Determine the Sign of in Each Interval
To identify the intervals where is increasing or decreasing, we examine the sign of in each subinterval:
Since is a product of and , the sign of depends on the sign of :
- In , (positive), so (negative), meaning is decreasing.
- In , (negative), so (positive), meaning is increasing.
- In , (positive), so (negative), meaning is decreasing.
Final Answer
The function is:
- Decreasing on and .
- Increasing on .
Would you like further details, or do you have any questions?
- How does the derivative of a trigonometric function determine intervals of increase or decrease?
- What is the impact of different constants in the argument of trigonometric functions on their period?
- How can we verify intervals of increase or decrease graphically?
- What is the relationship between critical points and intervals of monotonicity?
- How do changes in the domain affect the analysis of periodic functions?
Tip: Use the sign of the derivative to determine if a function is increasing or decreasing within specific intervals.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Trigonometry
Calculus
Derivative
Increasing and Decreasing Functions
Formulas
Derivative of cos(kx) is -k*sin(kx)
Set derivative equal to zero to find critical points
Theorems
First Derivative Test
Suitable Grade Level
Grades 10-12
Related Recommendation
Determining Increasing and Decreasing Intervals of F(x) = cos(x)
Interval Analysis of f(x) = x^3 / (2 - x^3)
Analyze Increasing and Decreasing Intervals of f(x) = 3cos²(x)
Determine Intervals of Increase and Decrease for f(x) = -2cos(x) - x on [0, π]
Determining Intervals of Increase and Decrease for a Function