Question
Identify the x
-values for the maximums, minimums, and zeros of y=− 4cos((1/3)x)
. Assume that n
is any integer.
The maximums occur when x
is
6πn
(3π/2)+6πn
3π+6πn
5π+6πn
The minimums occur when x
is
6πn
5π+6πn
(3π/2)+6πn
3π+6πn
The zeros occur when x
is
3πn
(2π/3)n
(3π/2)+3πn
(π/6)n
-values for the maximums, minimums, and zeros of y=− 4cos((1/3)x)
. Assume that n
is any integer.
The maximums occur when x
is
6πn
(3π/2)+6πn
3π+6πn
5π+6πn
The minimums occur when x
is
6πn
5π+6πn
(3π/2)+6πn
3π+6πn
The zeros occur when x
is
3πn
(2π/3)n
(3π/2)+3πn
(π/6)n
Answers
GPT-4o mini
To find the maximums, minimums, and zeros of the function \( y = -4\cos\left(\frac{1}{3}x\right) \), we can analyze the behavior of the cosine function.
### 1. Maximums:
The function \( -4\cos\left(\frac{1}{3}x\right) \) achieves its maximum value when the cosine function is at its minimum value of -1. This occurs when:
\[
\cos\left(\frac{1}{3}x\right) = -1
\]
The general solution for the cosine function being -1 is given by:
\[
\frac{1}{3}x = (2n+1)\pi \quad \Rightarrow \quad x = 3(2n+1)\pi = 6\pi n + 3\pi
\]
Thus, the maximums occur when \( x \) is given by:
\[
x = 6\pi n + 3\pi
\]
This can be rewritten as \( x = 6\pi n + 3\pi = 3\pi + 6\pi n \).
### 2. Minimums:
The function achieves its minimum value when the cosine function is at its maximum value of 1. This occurs when:
\[
\cos\left(\frac{1}{3}x\right) = 1
\]
The general solution for the cosine function being 1 is given by:
\[
\frac{1}{3}x = 2n\pi \quad \Rightarrow \quad x = 6n\pi
\]
Thus, the minimums occur when \( x \) is:
\[
x = 6\pi n
\]
### 3. Zeros:
The function \( y = -4\cos\left(\frac{1}{3}x\right) \) is equal to zero when:
\[
-4\cos\left(\frac{1}{3}x\right) = 0 \quad \Rightarrow \quad \cos\left(\frac{1}{3}x\right) = 0
\]
The cosine function is zero at:
\[
\frac{1}{3}x = \frac{\pi}{2} + n\pi \quad \Rightarrow \quad x = \left(\frac{\pi}{2} + n\pi\right) \cdot 3 = \frac{3\pi}{2} + 3n\pi
\]
Thus, the zeros occur when \( x \) is:
\[
x = \frac{3\pi}{2} + 3\pi n
\]
### Summary of results:
- **Maximums:** \( x = 3\pi + 6\pi n \)
- **Minimums:** \( x = 6\pi n \)
- **Zeros:** \( x = \frac{3\pi}{2} + 3\pi n \)
Now, comparing these results to your options:
- **Maximums occur when:** \( 3\pi + 6\pi n \)
- **Minimums occur when:** \( 6\pi n \)
- **Zeros occur when:** \( \frac{3\pi}{2} + 3\pi n \)
Let me know if you need any further assistance!
### 1. Maximums:
The function \( -4\cos\left(\frac{1}{3}x\right) \) achieves its maximum value when the cosine function is at its minimum value of -1. This occurs when:
\[
\cos\left(\frac{1}{3}x\right) = -1
\]
The general solution for the cosine function being -1 is given by:
\[
\frac{1}{3}x = (2n+1)\pi \quad \Rightarrow \quad x = 3(2n+1)\pi = 6\pi n + 3\pi
\]
Thus, the maximums occur when \( x \) is given by:
\[
x = 6\pi n + 3\pi
\]
This can be rewritten as \( x = 6\pi n + 3\pi = 3\pi + 6\pi n \).
### 2. Minimums:
The function achieves its minimum value when the cosine function is at its maximum value of 1. This occurs when:
\[
\cos\left(\frac{1}{3}x\right) = 1
\]
The general solution for the cosine function being 1 is given by:
\[
\frac{1}{3}x = 2n\pi \quad \Rightarrow \quad x = 6n\pi
\]
Thus, the minimums occur when \( x \) is:
\[
x = 6\pi n
\]
### 3. Zeros:
The function \( y = -4\cos\left(\frac{1}{3}x\right) \) is equal to zero when:
\[
-4\cos\left(\frac{1}{3}x\right) = 0 \quad \Rightarrow \quad \cos\left(\frac{1}{3}x\right) = 0
\]
The cosine function is zero at:
\[
\frac{1}{3}x = \frac{\pi}{2} + n\pi \quad \Rightarrow \quad x = \left(\frac{\pi}{2} + n\pi\right) \cdot 3 = \frac{3\pi}{2} + 3n\pi
\]
Thus, the zeros occur when \( x \) is:
\[
x = \frac{3\pi}{2} + 3\pi n
\]
### Summary of results:
- **Maximums:** \( x = 3\pi + 6\pi n \)
- **Minimums:** \( x = 6\pi n \)
- **Zeros:** \( x = \frac{3\pi}{2} + 3\pi n \)
Now, comparing these results to your options:
- **Maximums occur when:** \( 3\pi + 6\pi n \)
- **Minimums occur when:** \( 6\pi n \)
- **Zeros occur when:** \( \frac{3\pi}{2} + 3\pi n \)
Let me know if you need any further assistance!