Find the maximum and minimum value of `3 sin 2x+4 cos 2x+3`.

Answer

To find the maximum and minimum values of the function \( f(x) = 3 \sin 2x + 4 \cos 2x + 3 \), we can follow these steps:

### Step 1: Identify the function
We start with the function:
\[ f(x) = 3 \sin 2x + 4 \cos 2x + 3 \]
We can separate the trigonometric part from the constant:
\[ h(x) = 3 \sin 2x + 4 \cos 2x \]

### Step 2: Find the maximum and minimum of \( h(x) \)
The expression \( h(x) = a \sin x + b \cos x \) has a known maximum and minimum value, which can be calculated using the formula:
- Maximum value: \( \sqrt{a^2 + b^2} \)
- Minimum value: \( -\sqrt{a^2 + b^2} \)

Here, \( a = 3 \) and \( b = 4 \).

### Step 3: Calculate \( \sqrt{a^2 + b^2} \)
Now we compute:
\[ \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Determine the maximum and minimum values of \( h(x) \)
From the calculations:
- Maximum value of \( h(x) = 5 \)
- Minimum value of \( h(x) = -5 \)

### Step 5: Adjust for the constant in \( f(x) \)
Now, we need to add the constant \( 3 \) back to the maximum and minimum values of \( h(x) \):
- Maximum value of \( f(x) = 5 + 3 = 8 \)
- Minimum value of \( f(x) = -5 + 3 = -2 \)

### Conclusion
Thus, the maximum value of \( f(x) \) is \( 8 \) and the minimum value is \( -2 \).

### Final Answer
Maximum value: \( 8 \)
Minimum value: \( -2 \)