Question

Identify the x-values for the maximums, minimums, and zeros of y=5sin(5x)
. Assume that n
is any integer.



The maximums occur when x
is



(1/10)π+(2/5)πn
(2/5)πn
(3/10)π+(2/5)πn
(1/5)π+(2/5)πn




The minimums occur when x
is



(1/5)π+(2/5)πn
(2/5)πn
(3/10)π+(2/5)πn
(1/10)π+(2/5)πn




The zeros occur when x
is



(1/10)πn
(1/5)πn
(2/5)πn
(1/10)π+(1/5)πn

Answer 1

To analyze the function \( y = 5\sin(5x) \), we need to identify where the maximums, minimums, and zeros occur.

1. **Maximums**: The sine function \( \sin(5x) \) reaches its maximum value of 1. Thus, \( 5\sin(5x) \) reaches its maximum of \( 5 \) when:
\[
5x = \frac{\pi}{2} + 2k\pi \quad (k \text{ is an integer})
\]
Rearranging gives:
\[
x = \frac{1}{10}\pi + \frac{2}{5}\pi k
\]
Therefore, the maximums occur when \( x \) is:
\[
\frac{1}{10}\pi + \frac{2}{5}\pi n
\]

2. **Minimums**: The sine function \( \sin(5x) \) reaches its minimum value of -1. Thus, \( 5\sin(5x) \) reaches its minimum of \( -5 \) when:
\[
5x = \frac{3\pi}{2} + 2k\pi \quad (k \text{ is an integer})
\]
Rearranging gives:
\[
x = \frac{3}{10}\pi + \frac{2}{5}\pi k
\]
Therefore, the minimums occur when \( x \) is:
\[
\frac{3}{10}\pi + \frac{2}{5}\pi n
\]

3. **Zeros**: The sine function \( \sin(5x) \) is zero when:
\[
5x = k\pi \quad (k \text{ is an integer})
\]
Rearranging gives:
\[
x = \frac{1}{5}\pi k
\]
Therefore, the zeros occur when \( x \) is:
\[
\frac{1}{5}\pi n
\]

### Summary
- The maximums occur when \( x \) is \( \frac{1}{10}\pi + \frac{2}{5}\pi n \).
- The minimums occur when \( x \) is \( \frac{3}{10}\pi + \frac{2}{5}\pi n \).
- The zeros occur when \( x \) is \( \frac{1}{5}\pi n \).