Find the maximum or minimum values, if any, of the following funcitons without using the derivatives :
`f(x)=|sin4x+3|.`

Answer

To find the maximum and minimum values of the function \( f(x) = |\sin(4x) + 3| \) without using derivatives, we can analyze the function step by step.

### Step 1: Understand the sine function
The sine function, \( \sin(4x) \), oscillates between -1 and 1 for all values of \( x \). Therefore, we can express this mathematically as:
\[ -1 \leq \sin(4x) \leq 1 \]

### Step 2: Shift the sine function
Next, we add 3 to the sine function:
\[ -1 + 3 \leq \sin(4x) + 3 \leq 1 + 3 \]
This simplifies to:
\[ 2 \leq \sin(4x) + 3 \leq 4 \]

### Step 3: Apply the absolute value
Now, we need to consider the absolute value of \( \sin(4x) + 3 \):
\[ f(x) = |\sin(4x) + 3| \]
Since \( \sin(4x) + 3 \) is always positive (it ranges from 2 to 4), we can simplify the absolute value:
\[ f(x) = \sin(4x) + 3 \]

### Step 4: Determine the range of \( f(x) \)
From our previous step, we know:
\[ 2 \leq \sin(4x) + 3 \leq 4 \]
Thus, the function \( f(x) \) will take values between 2 and 4.

### Conclusion
The minimum value of \( f(x) \) is 2, and the maximum value of \( f(x) \) is 4.

**Final Answer:**
- Minimum value of \( f(x) \) is 2.
- Maximum value of \( f(x) \) is 4.