Math Problem Statement
Solution
Let's solve the problem based on the image you uploaded, which concerns Rolle's theorem applied to the function on the interval .
Step 1: Check if Rolle's theorem can be applied.
For Rolle's theorem to apply, three conditions must be met:
-
Continuity on the closed interval :
The function is continuous on because the sine function is continuous everywhere. -
Differentiability on the open interval :
The function is differentiable on because the sine function is differentiable everywhere. -
:
Here, and . Thus, , so this condition is also satisfied.
Since all three conditions are satisfied, Rolle's theorem can be applied.
Step 2: Find all values of in the open interval such that .
To do this, we first need to compute the derivative of : Now, we set to find critical points: The cosine function equals 0 at: in the interval .
Thus, the values of where are:
Final Answer:
- Rolle's theorem can be applied.
- The values of are .
Would you like more details or have any questions?
Here are 5 related questions for further practice:
- What is the geometric interpretation of Rolle's theorem?
- If , can Rolle's theorem be applied on the interval ?
- What is the derivative of , and when does it equal zero on ?
- How does Rolle's theorem differ from the Mean Value Theorem?
- What are the conditions for applying the Mean Value Theorem?
Tip: Remember that the function must satisfy all three conditions of Rolle's theorem before you can apply it.
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Trigonometry
Formulas
Derivative of sin(x): cos(x)
f'(x) = 2cos(x)
Theorems
Rolle's theorem
Suitable Grade Level
Grades 11-12