Math Problem Statement

Determine whether Rolle's theorem can be applied to f on the closed interval [a, b]. f(x) = 2sin(x), [0, 2π]. Select all that apply. If Rolle's theorem can be applied, find all values of c in the open interval (a, b) such that f'(c) = 0.

Solution

Let's solve the problem based on the image you uploaded, which concerns Rolle's theorem applied to the function f(x)=2sin(x)f(x) = 2 \sin(x) on the interval [0,2π][0, 2\pi].

Step 1: Check if Rolle's theorem can be applied.

For Rolle's theorem to apply, three conditions must be met:

  1. Continuity on the closed interval [a,b][a, b]:
    The function f(x)=2sin(x)f(x) = 2 \sin(x) is continuous on [0,2π][0, 2\pi] because the sine function is continuous everywhere.

  2. Differentiability on the open interval (a,b)(a, b):
    The function f(x)=2sin(x)f(x) = 2 \sin(x) is differentiable on (0,2π)(0, 2\pi) because the sine function is differentiable everywhere.

  3. f(a)=f(b)f(a) = f(b):
    Here, f(0)=2sin(0)=0f(0) = 2 \sin(0) = 0 and f(2π)=2sin(2π)=0f(2\pi) = 2 \sin(2\pi) = 0. Thus, f(0)=f(2π)f(0) = f(2\pi), so this condition is also satisfied.

Since all three conditions are satisfied, Rolle's theorem can be applied.

Step 2: Find all values of cc in the open interval (0,2π)(0, 2\pi) such that f(c)=0f'(c) = 0.

To do this, we first need to compute the derivative of f(x)f(x): f(x)=ddx[2sin(x)]=2cos(x).f'(x) = \frac{d}{dx}[2 \sin(x)] = 2 \cos(x). Now, we set f(x)=0f'(x) = 0 to find critical points: 2cos(x)=0    cos(x)=0.2 \cos(x) = 0 \implies \cos(x) = 0. The cosine function equals 0 at: x=π2,3π2x = \frac{\pi}{2}, \frac{3\pi}{2} in the interval (0,2π)(0, 2\pi).

Thus, the values of cc where f(c)=0f'(c) = 0 are: c=π2,3π2.c = \frac{\pi}{2}, \frac{3\pi}{2}.

Final Answer:

  • Rolle's theorem can be applied.
  • The values of cc are π2,3π2\frac{\pi}{2}, \frac{3\pi}{2}.

Would you like more details or have any questions?

Here are 5 related questions for further practice:

  1. What is the geometric interpretation of Rolle's theorem?
  2. If f(x)=x2f(x) = x^2, can Rolle's theorem be applied on the interval [1,1][-1, 1]?
  3. What is the derivative of f(x)=3sin(x)f(x) = 3\sin(x), and when does it equal zero on [0,2π][0, 2\pi]?
  4. How does Rolle's theorem differ from the Mean Value Theorem?
  5. 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