Question

Determine the minimum and maximum value of the following trigonometric function.
f, of, x, equals, minus, 3, sine, 2, x, plus, 3
f(x)=−3sin2x+3

Answer 1

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 \( f(x) \).

1. **Understanding the sine function**:
The sine function, \( \sin(2x) \), oscillates between -1 and 1. Therefore,
\[
-1 \leq \sin(2x) \leq 1.
\]

2. **Calculating the values of \( f(x) \)**:
Now we will substitute the extreme values of \( \sin(2x) \) into the function \( f(x) \):

- When \( \sin(2x) = 1 \):
\[
f(x) = -3(1) + 3 = -3 + 3 = 0.
\]

- When \( \sin(2x) = -1 \):
\[
f(x) = -3(-1) + 3 = 3 + 3 = 6.
\]

3. **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**: \( 6 \)