At which [tex]x[/tex]-value does the cosine function attain a maximum value?

A. [tex]x=0[/tex]
B. [tex]x=\frac{\pi}{2}[/tex]
C. [tex]x=1[/tex]
D. [tex]x=\frac{3 \pi}{2}[/tex]



Answer :

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].