To determine the formula that gives the [tex]\( x \)[/tex]-coordinates of the maximum values for the function [tex]\( y = \cos(x) \)[/tex], we need to identify where the cosine function reaches its peak value of 1.
The cosine function [tex]\(\cos(x)\)[/tex] has a period of [tex]\(2\pi\)[/tex], meaning it repeats its values every [tex]\(2\pi\)[/tex] units. The function [tex]\(\cos(x)\)[/tex] reaches its maximum value of 1 at multiple points across its domain.
Let's consider the general behavior of the cosine function:
- The cosine function reaches its maximum value of 1 at [tex]\( x = 0 \)[/tex].
- Because the function is periodic with a period of [tex]\(2\pi\)[/tex], it reaches the same maximum value at [tex]\( x = 2\pi, 4\pi, 6\pi, \ldots \)[/tex].
- Similarly, it reaches the same maximum value at [tex]\( x = -2\pi, -4\pi, -6\pi, \ldots \)[/tex].
The common characteristic among these [tex]\( x \)[/tex]-values is that they can be represented in the form [tex]\( x = k\pi \)[/tex], where [tex]\( k \)[/tex] is an even integer (including zero).
Thus, the correct answer is:
[tex]\[ \text{\boldmath $k \pi$ for any integer $k$} \][/tex]
This formula gives the [tex]\( x \)[/tex]-coordinates where the cosine function reaches its maximum value.
