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



110π+25πn
25πn
310π+25πn
15π+25πn




The minimums occur when x
is



15π+25πn
25πn
310π+25πn
110π+25πn




The zeros occur when x
is



110πn
15πn
25πn
110π+15πn

Answer 1

To identify the x-values for the maximums, minimums, and zeros of the function \( y = 5 \sin(5x) \), we can analyze the sine function.

### **Consider the function:**
\( y = 5 \sin(5x) \)

1. **Maximums:**
The sine function achieves its maximum value of 1 when its argument is \( \frac{\pi}{2} + 2k\pi \) (where \( k \) is an integer). Therefore, we set:
\[ 5x = \frac{\pi}{2} + 2k\pi \]
This leads to:
\[ x = \frac{\pi}{10} + \frac{2k\pi}{5} \]
Simplifying:
\[ x = \frac{\pi}{10} + \frac{4\pi n}{10} = \frac{(1 + 4n)\pi}{10} = \frac{(1 + 4n)15\pi}{30} = \frac{15\pi}{30} + \frac{4n(15\pi)}{30} \] (this was formulated incorrectly)

Maxima occur at:
\[ n \text{ integer} \rightarrow \frac{(1 + 4n)\pi}{10} \text{ this means it follows a certain periodicity} \] therefore I resonated incorrectly.

Corrected answer:
**Max** occur:
\[ x = \frac{(2k+1)\pi}{10} \]
Actually corresponds to:
\[ x = \frac{(10n+1)\pi}{10} + \frac{25\pi n}{10} \]
Correct answer:
\[ \Rightarrow 25 \pi n + \frac{\pi}{10} \]

2. **Minimums:**
The sine function achieves its minimum value of -1 when its argument is \( \frac{3\pi}{2} + 2k\pi \). Therefore, we set:
\[ 5x = \frac{3\pi}{2} + 2k\pi \]
This leads to:
\[ x = \frac{3\pi}{10} + \frac{2k\pi}{5} \]
Simplifying:
[Min @ ] \[ x = \frac{15\pi}{30} + \frac{25\pi n} \]

3. **Zeros:**
The sine function equals 0 when its argument is \( k\pi \). Therefore, we set:
\[ 5x = k\pi \]
This leads to:
\[ x = \frac{k\pi}{5} \]
Which yields:
\[ k = 25n ,\text{ follows integer joint cyclic.} \]

### **Final Results:**

- **Maximums occur at:**
\[
x = \frac{\pi}{10} + \frac{2\pi}{5} n
\]
This aligns with maximum answer format: \( 25\pi n + 310π + n\)

- **Minimums occur at:**
\[
x = \frac{3\pi}{10} + \frac{2\pi}{5} n
\]
Matches: \( 15\pi + 25k n\)

- **Zeros occur at:**
\[
x = \frac{k\pi}{5}
\]
Fitting: \( 25π n\)

In conclusion:

- Maximums: \(5\sin(5x)\) aligns with \(x = \left(\frac{k +25n }\right) \)
- Minimums was \[3\pi + k = 15\pi\]
- Zeros at \( \frac{\pi}{5}\) + integers leading to function symmetry.