Find the maximum or minimum values, if any, of the following funcitons without using the derivatives :
`f(x)=sin2x+5`

Answer

To find the maximum and minimum values of the function \( f(x) = \sin(2x) + 5 \) without using derivatives, we can analyze the properties of the sine function.

### Step-by-Step Solution:

1. **Understand the Range of the Sine Function**:
The sine function, \( \sin(\theta) \), oscillates between -1 and 1 for all values of \( \theta \). Therefore, we have:
\[ -1 \leq \sin(2x) \leq 1 \]

2. **Shift the Range**:
Since our function is \( f(x) = \sin(2x) + 5 \), we can shift the entire range of the sine function by adding 5:
\[ -1 + 5 \leq \sin(2x) + 5 \leq 1 + 5 \]
This simplifies to:
\[ 4 \leq f(x) \leq 6 \]

3. **Identify the Maximum and Minimum Values**:
From the inequality \( 4 \leq f(x) \leq 6 \), we can conclude:
- The **minimum value** of \( f(x) \) is **4**.
- The **maximum value** of \( f(x) \) is **6**.

### Conclusion:
Thus, the minimum value of the function \( f(x) = \sin(2x) + 5 \) is **4**, and the maximum value is **6**.