To determine at which [tex]\( x \)[/tex]-value the cosine function attains its maximum value, let's evaluate the cosine function at the given [tex]\( x \)[/tex]-values:
1. At [tex]\( x = 0 \)[/tex]:
[tex]\[
\cos(0) = 1
\][/tex]
2. At [tex]\( x = \frac{\pi}{2} \)[/tex]:
[tex]\[
\cos\left(\frac{\pi}{2}\right) \approx 6.123233995736766 \times 10^{-17} \approx 0
\][/tex]
3. At [tex]\( x = 1 \)[/tex]:
[tex]\[
\cos(1) \approx 0.5403023058681398
\][/tex]
4. At [tex]\( x = \frac{3\pi}{2} \)[/tex]:
[tex]\[
\cos\left(\frac{3\pi}{2}\right) \approx -1.8369701987210297 \times 10^{-16} \approx 0
\][/tex]
Comparing these values, we get:
- [tex]\(\cos(0) = 1\)[/tex]
- [tex]\(\cos\left(\frac{\pi}{2}\right) \approx 0\)[/tex]
- [tex]\(\cos(1) \approx 0.54\)[/tex]
- [tex]\(\cos\left(\frac{3\pi}{2}\right) \approx 0\)[/tex]
From these evaluations, it is clear that the maximum value of the cosine function from the given [tex]\( x \)[/tex]-values is [tex]\( 1 \)[/tex], which occurs at [tex]\( x = 0 \)[/tex].
Thus, the cosine function attains its maximum value at [tex]\( x = 0 \)[/tex].
