Questions: Answer the following questions about the function whose derivative is given below. a. What are the critical points of f ? b. On what open intervals is f increasing or decreasing? c. At what points, if any, does f assume local maximum or minimum values? f'(x)=(sqrt(3) sin x+cos x)(sin x-sqrt(3) cos x), 0 ≤ x ≤ 2 pi a. What are the critical points of f ? Select the correct choice below and, if necessary, fill in the answer box to A. The critical point(s) of f is/are x= (Type an exact answer, using pi as needed. Use a comma to separate answers as needed.) B. The function f has no critical points.

Answer the following questions about the function whose derivative is given below.
a. What are the critical points of f ?
b. On what open intervals is f increasing or decreasing?
c. At what points, if any, does f assume local maximum or minimum values?

f'(x)=(sqrt(3) sin x+cos x)(sin x-sqrt(3) cos x), 0 ≤ x ≤ 2 pi

a. What are the critical points of f ? Select the correct choice below and, if necessary, fill in the answer box to
A. The critical point(s) of f is/are x=
(Type an exact answer, using pi as needed. Use a comma to separate answers as needed.)
B. The function f has no critical points.

Transcript text: Answer the following questions about the function whose derivative is given below. a. What are the critical points of $f$ ? b. On what open intervals is $f$ increasing or decreasing? c. At what points, if any, does f assume local maximum or minimum values? \[ f^{\prime}(x)=(\sqrt{3} \sin x+\cos x)(\sin x-\sqrt{3} \cos x), 0 \leq x \leq 2 \pi \] a. What are the critical points of $f$ ? Select the correct choice below and, if necessary, fill in the answer box to A. The critical point(s) of $f$ is/are $x=$ $\square$ (Type an exact answer, using $\pi$ as needed. Use a comma to separate answers as needed.) B. The function $f$ has no critical points.

Solution

Solution Steps

To solve the given problem, we need to find the critical points of the function $ f $ whose derivative is given. Critical points occur where the derivative is zero or undefined. We will set the given derivative equal to zero and solve for $ x $ within the interval $ [0, 2\pi] $.

Solution Approach

Set the derivative $ f'(x) = (\sqrt{3} \sin x + \cos x)(\sin x - \sqrt{3} \cos x) $ equal to zero.
Solve the resulting equations $ \sqrt{3} \sin x + \cos x = 0 $ and $ \sin x - \sqrt{3} \cos x = 0 $ for $ x $ within the interval $ [0, 2\pi] $.
Combine the solutions to find all critical points.

Step 1: Finding Critical Points

To find the critical points of the function $ f $, we set the derivative $ f'(x) = (\sqrt{3} \sin x + \cos x)(\sin x - \sqrt{3} \cos x) $ equal to zero. Solving this equation, we find that the critical point occurs at:

\[ x \approx 1.0472 \]

Step 2: Expressing Critical Points

The critical point can be expressed in terms of radians. Notably, $ 1.0472 $ radians is equivalent to $ \frac{\pi}{3} $. Thus, the critical point is:

\[ x = \frac{\pi}{3} \]