Sure, let's find out at what times the pendulum's movement will be 4 meters from its starting position using the given equation: [tex]\( f(x) = -2 \cos \left(\frac{x}{4}\right) + 2 \)[/tex].
1. Set the equation equal to 4: To determine the times at which the displacement is 4 meters, we set up the equation:
[tex]\[
-2 \cos \left(\frac{x}{4}\right) + 2 = 4
\][/tex]
2. Isolate the cosine term: Subtract 2 from both sides to isolate the cosine term:
[tex]\[
-2 \cos \left(\frac{x}{4}\right) = 2
\][/tex]
3. Divide by -2: Simplify by dividing both sides by -2:
[tex]\[
\cos \left(\frac{x}{4}\right) = -1
\][/tex]
4. Solve for [tex]\(\frac{x}{4}\)[/tex]: We know that [tex]\(\cos(\theta) = -1\)[/tex] happens when the angle [tex]\(\theta\)[/tex] is an odd multiple of [tex]\(\pi\)[/tex]:
[tex]\[
\frac{x}{4} = (2k + 1) \pi
\][/tex]
where [tex]\(k\)[/tex] is an integer (representing the periodic nature of the cosine function).
5. Solve for [tex]\(x\)[/tex]: Multiply both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 4(2k + 1) \pi
\][/tex]
6. General solution: We can rewrite this solution to express it more clearly:
[tex]\[
x = 4\pi (2k + 1)
\][/tex]
which simplifies to
[tex]\[
x = 4\pi + 8k\pi
\][/tex]
Here, for simplicity, we can represent [tex]\(k = n\)[/tex], where [tex]\(n\)[/tex] is any integer:
[tex]\[
x = 4\pi + 4n\pi
\][/tex]
Therefore, the times when the pendulum's movement will be 4 meters from its starting position are given by:
[tex]\[
\boxed{4\pi + 4n \text{ seconds}}
\][/tex]
Thus, the correct answer is [tex]\(4\pi + 4n \)[/tex] seconds.
