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.
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Understanding the sine component: The sine function \( \sin(4x) \) oscillates between \(-1\) and \(1\).
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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: \[ \text{Maximum of } 4\sin(4x) = 4 \cdot 1 = 4 \] \[ \text{Minimum of } 4\sin(4x) = 4 \cdot (-1) = -4 \]
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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 \( f(x) \): \[ f_{\text{min}} = -4 - 1 = -5 \]
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Conclusion:
- The maximum value of \( f(x) \) is \(3\).
- The minimum value of \( f(x) \) is \(-5\).
Thus, the final result is: \[ \text{Maximum value: } 3, \quad \text{Minimum value: } -5 \]