Question

Determine the minimum and maximum value of the following trigonometric function.
f, of, x, equals, 4, cosine, 3, x, plus, 3
f(x)=4cos3x+3

Answer 1

To find the minimum and maximum values of the function \( f(x) = 4 \cos(3x) + 3 \), we first analyze the cosine component.

1. **Understand the cosine function**: The function \( \cos(3x) \) oscillates between -1 and 1 for all values of \( x \).

2. **Apply this to the function**:
\[
f(x) = 4 \cos(3x) + 3
\]
Let's find the extreme values of \( f(x) \) by evaluating the extreme values of \( \cos(3x) \):
- The maximum value of \( \cos(3x) \) is 1.
- The minimum value of \( \cos(3x) \) is -1.

3. **Calculate the max and min of \( f(x) \)**:
- **Maximum value**:
\[
f_{\text{max}} = 4(1) + 3 = 4 + 3 = 7
\]

- **Minimum value**:
\[
f_{\text{min}} = 4(-1) + 3 = -4 + 3 = -1
\]

4. **Conclusion**: Thus, the minimum and maximum values of the function \( f(x) = 4 \cos(3x) + 3 \) are:
- Minimum value: \( -1 \)
- Maximum value: \( 7 \)