Question
Determine the minimum and maximum value of the following trigonometric function.
f, of, x, equals, minus, 5, cosine, 4, x
f(x)=−5cos4x
f, of, x, equals, minus, 5, cosine, 4, x
f(x)=−5cos4x
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Answered by
GPT-4o mini
To determine the minimum and maximum values of the function \( f(x) = -5 \cos(4x) \), we first need to analyze the behavior of the cosine function.
The cosine function, \( \cos(4x) \), oscillates between -1 and 1. Therefore, we can find the maximum and minimum values of \( f(x) \) by substituting these values into the function.
1. **Maximum Value**:
- When \( \cos(4x) = -1 \):
\[
f(x) = -5 \times (-1) = 5
\]
2. **Minimum Value**:
- When \( \cos(4x) = 1 \):
\[
f(x) = -5 \times 1 = -5
\]
Thus, the maximum value of the function \( f(x) = -5 \cos(4x) \) is **5**, and the minimum value is **-5**.
In summary:
- **Maximum value**: \( 5 \)
- **Minimum value**: \( -5 \)
The cosine function, \( \cos(4x) \), oscillates between -1 and 1. Therefore, we can find the maximum and minimum values of \( f(x) \) by substituting these values into the function.
1. **Maximum Value**:
- When \( \cos(4x) = -1 \):
\[
f(x) = -5 \times (-1) = 5
\]
2. **Minimum Value**:
- When \( \cos(4x) = 1 \):
\[
f(x) = -5 \times 1 = -5
\]
Thus, the maximum value of the function \( f(x) = -5 \cos(4x) \) is **5**, and the minimum value is **-5**.
In summary:
- **Maximum value**: \( 5 \)
- **Minimum value**: \( -5 \)
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