Question

Determine the minimum and maximum value of the following trigonometric function.
f, of, x, equals, 4, sine, 4, x, minus, 1
f(x)=4sin4x−1

Answer 1

To determine the minimum and maximum values of the function \( f(x) = 4\sin(4x) - 1 \), we need to analyze the behavior of the sine function.

1. **Understanding the sine component**: The sine function \( \sin(4x) \) oscillates between \(-1\) and \(1\).

2. **Finding the maximum and minimum of \(4\sin(4x)\)**:
- The maximum value of \( \sin(4x) \) is \(1\).
- The minimum value of \( \sin(4x) \) is \(-1\).
- Therefore:
\[
\text{Maximum of } 4\sin(4x) = 4 \cdot 1 = 4
\]
\[
\text{Minimum of } 4\sin(4x) = 4 \cdot (-1) = -4
\]

3. **Adjusting for the constant term \(-1\)**:
- To find the corresponding maximum value of \( f(x) \):
\[
f_{\text{max}} = 4 - 1 = 3
\]
- To find the corresponding minimum value of \( f(x) \):
\[
f_{\text{min}} = -4 - 1 = -5
\]

4. **Conclusion**:
- The maximum value of \( f(x) \) is \(3\).
- The minimum value of \( f(x) \) is \(-5\).

Thus, the final result is:
\[
\text{Maximum value: } 3, \quad \text{Minimum value: } -5
\]