To determine the minimum and maximum values of the function \( f(x) = -3 \sin(2x) + 3 \), we can analyze the range of the sine function and how it affects \( f(x) \).
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Understanding the sine function: The sine function, \( \sin(2x) \), oscillates between -1 and 1. Therefore, \[ -1 \leq \sin(2x) \leq 1. \]
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Calculating the values of \( f(x) \): Now we will substitute the extreme values of \( \sin(2x) \) into the function \( f(x) \):
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When \( \sin(2x) = 1 \): \[ f(x) = -3(1) + 3 = -3 + 3 = 0. \]
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When \( \sin(2x) = -1 \): \[ f(x) = -3(-1) + 3 = 3 + 3 = 6. \]
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Identifying the minimum and maximum values: From the calculations, we can summarize:
- The minimum value occurs when \( \sin(2x) = 1 \), resulting in \( f(x) = 0 \).
- The maximum value occurs when \( \sin(2x) = -1 \), resulting in \( f(x) = 6 \).
Thus, the minimum and maximum values of the function \( f(x) = -3 \sin(2x) + 3 \) are:
- Minimum value: \( 0 \)
- Maximum value: \( 6 \)
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