The maximum and minimum values of function `f(x)=sin 3x+4` are respectively:

Options

A. 5 and 3
B. 6 and 4
C. 4 and 3
D. None of these.

Answer

To find the maximum and minimum values of the function \( f(x) = \sin(3x) + 4 \), we can follow these steps:

### Step 1: Understand the Range of the Sine Function
The sine function, \( \sin(\theta) \), varies between -1 and 1 for any angle \( \theta \). Therefore, we can express this as:
\[ -1 \leq \sin(3x) \leq 1 \]

### Step 2: Transform the Range for \( f(x) \)
Since our function is \( f(x) = \sin(3x) + 4 \), we can add 4 to each part of the inequality:
\[ -1 + 4 \leq \sin(3x) + 4 \leq 1 + 4 \]
This simplifies to:
\[ 3 \leq f(x) \leq 5 \]

### Step 3: Identify Maximum and Minimum Values
From the inequality \( 3 \leq f(x) \leq 5 \), we can determine:
- The **minimum value** of \( f(x) \) is **3**.
- The **maximum value** of \( f(x) \) is **5**.

### Conclusion
Thus, the maximum and minimum values of the function \( f(x) = \sin(3x) + 4 \) are:
- Maximum value: **5**
- Minimum value: **3**