To find the [tex]\(x\)[/tex]-intercepts of the function [tex]\(y = \cot(3x)\)[/tex], we need to determine where [tex]\(\cot(3x) = 0\)[/tex].
The cotangent function [tex]\(\cot(x)\)[/tex] is zero at specific points, which are given by:
[tex]\[ \cot(x) = 0 \iff x = \frac{\pi}{2} + k\pi \][/tex]
where [tex]\(k\)[/tex] is any integer.
For the function [tex]\(y = \cot(3x)\)[/tex], we modify these points accordingly:
[tex]\[ 3x = \frac{\pi}{2} + k\pi \][/tex]
To solve for [tex]\(x\)[/tex], we divide the entire equation by 3:
[tex]\[ x = \frac{\pi}{6} + \frac{k\pi}{3} \][/tex]
Thus, the [tex]\(x\)[/tex]-intercepts of [tex]\(y = \cot(3x)\)[/tex] occur at:
[tex]\[ x = \frac{\pi}{6} + \frac{k\pi}{3} \][/tex]
Now we need to check which one of the given points satisfies this condition:
### Checking [tex]\(\left(\frac{\pi}{8}, 0\right)\)[/tex]
Substitute [tex]\(x = \frac{\pi}{8}\)[/tex]:
[tex]\[ 3 \left(\frac{\pi}{8}\right) = \frac{3\pi}{8} \][/tex]
To see if this can be written in the form [tex]\(\frac{\pi}{2} + k\pi\)[/tex]:
[tex]\[ \frac{3\pi}{8} \neq \frac{\pi}{2} + k\pi \][/tex]
Thus, [tex]\(\left(\frac{\pi}{8}, 0\right)\)[/tex] is not an [tex]\(x\)[/tex]-intercept.
### Checking [tex]\(\left(\frac{\pi}{3}, 0\right)\)[/tex]
Substitute [tex]\(x = \frac{\pi}{3}\)[/tex]:
[tex]\[ 3 \left(\frac{\pi}{3}\right) = \pi \][/tex]
To see if this can be written in the form [tex]\(\frac{\pi}{2} + k\pi\)[/tex]:
[tex]\[ \pi = \frac{\pi}{2} + \pi \left( \text{for } k = 1 \right) \][/tex]
Thus, [tex]\(\left(\frac{\pi}{3}, 0\right)\)[/tex] is an [tex]\(x\)[/tex]-intercept.
### Checking [tex]\((3=0)\)[/tex]
This format is not a valid point for intercepts, so we can dismiss this immediately.
### Checking [tex]\((6 \pi, 0)\)[/tex]
Substitute [tex]\(x = 6\pi\)[/tex]:
[tex]\[ 3 \left(6\pi\right) = 18\pi \][/tex]
To see if this can be written in the form [tex]\(\frac{\pi}{2} + k\pi\)[/tex]:
[tex]\[ 18\pi = \frac{\pi}{2} + 35\pi \left( \text{for } k = 35 \right) \][/tex]
Thus, [tex]\((6 \pi, 0)\)[/tex] can be an [tex]\(x\)[/tex]-intercept, but it is much larger and doesn't fit in the standard interval check.
Based on this analysis, the valid [tex]\(x\)[/tex]-intercept closest and simplest is:
[tex]\(\boxed{\left( \frac{\pi}{3}, 0 \right)}\)[/tex]
