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Which formula gives the [tex]$x$[/tex]-coordinates of the maximum values for [tex]$y=\cos (x)$[/tex]?

A. [tex]$k \pi$[/tex] for any integer [tex]$k$[/tex]
B. [tex]$k \pi$[/tex] for [tex]$k=0, \pm 2, \pm 4, \ldots$[/tex]
C. [tex]$\frac{k \pi}{2}$[/tex] for any positive integer [tex]$k$[/tex]
D. [tex]$\frac{k \pi}{2}$[/tex] for [tex]$k=0, \pm 2, \pm 4, \ldots$[/tex]

Sagot :

GinnyAnswer
To determine the [tex]$x$[/tex]-coordinates where [tex]\( y = \cos(x) \)[/tex] reaches its maximum values, we need to consider the properties of the cosine function. The cosine function, [tex]\( \cos(x) \)[/tex], has a maximum value of 1 which occurs at specific points along the x-axis.

The cosine function achieves its maximum value at [tex]\( x = 2k\pi \)[/tex] where [tex]\( k \)[/tex] is an integer. This is because the cosine function has a period of [tex]\( 2\pi \)[/tex], meaning it repeats its values every [tex]\( 2\pi \)[/tex] interval. This makes the maximum points occur at:
[tex]\[ x = 2k\pi \][/tex]
for [tex]\( k = 0, \pm 1, \pm 2, \pm 3, \ldots \)[/tex].

Let's verify this by listing some values considering different integer values for [tex]\( k \)[/tex]:

1. For [tex]\( k = -5 \)[/tex]:
[tex]\[ x = 2 \cdot (-5) \cdot \pi = -10\pi \approx -31.41592653589793 \][/tex]

2. For [tex]\( k = -4 \)[/tex]:
[tex]\[ x = 2 \cdot (-4) \cdot \pi = -8\pi \approx -25.132741228718345 \][/tex]

3. For [tex]\( k = -3 \)[/tex]:
[tex]\[ x = 2 \cdot (-3) \cdot \pi = -6\pi \approx -18.84955592153876 \][/tex]

4. For [tex]\( k = -2 \)[/tex]:
[tex]\[ x = 2 \cdot (-2) \cdot \pi = -4\pi \approx -12.566370614359172 \][/tex]

5. For [tex]\( k = -1 \)[/tex]:
[tex]\[ x = 2 \cdot (-1) \cdot \pi = -2\pi \approx -6.283185307179586 \][/tex]

6. For [tex]\( k = 0 \)[/tex]:
[tex]\[ x = 2 \cdot 0 \cdot \pi = 0 \][/tex]

7. For [tex]\( k = 1 \)[/tex]:
[tex]\[ x = 2 \cdot 1 \cdot \pi = 2\pi \approx 6.283185307179586 \][/tex]

8. For [tex]\( k = 2 \)[/tex]:
[tex]\[ x = 2 \cdot 2 \cdot \pi = 4\pi \approx 12.566370614359172 \][/tex]

9. For [tex]\( k = 3 \)[/tex]:
[tex]\[ x = 2\cdot 3 \cdot \pi = 6\pi \approx 18.84955592153876 \][/tex]

10. For [tex]\( k = 4 \)[/tex]:
[tex]\[ x = 2 \cdot 4 \cdot \pi = 8\pi \approx 25.132741228718345 \][/tex]

11. For [tex]\( k = 5 \)[/tex]:
[tex]\[ x = 2 \cdot 5 \cdot \pi = 10\pi \approx 31.41592653589793 \][/tex]

From these values, we see the sequence of maximum points matches the result from running the given Python code. Hence, the values correspond to the formula [tex]\( x = 2k\pi \)[/tex] where [tex]\( k \)[/tex] is an integer.

Therefore, the correct answer is:
[tex]\[ \textbf{k} \pi \textbf{ for } k=0, \pm 2, \pm 4, \ldots \][/tex]
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