Determine whether the following function is increasing or decreasing in the given interval : `f(x) =cos (2x+(pi)/(4)), (3pi)/(8) le x le (5pi)/(8)`.

Answer

To determine whether the function \( f(x) = \cos\left(2x + \frac{\pi}{4}\right) \) is increasing or decreasing in the interval \( \left[\frac{3\pi}{8}, \frac{5\pi}{8}\right] \), we will follow these steps:

### Step 1: Find the derivative of the function
To analyze whether the function is increasing or decreasing, we first need to find the derivative of \( f(x) \).

\[ f'(x) = \frac{d}{dx} \left( \cos\left(2x + \frac{\pi}{4}\right)\right) \]

Using the chain rule, we have:

\[ f'(x) = -\sin\left(2x + \frac{\pi}{4}\right) \cdot \frac{d}{dx}(2x + \frac{\pi}{4}) = -\sin\left(2x + \frac{\pi}{4}\right) \cdot 2 \]

Thus,

\[ f'(x) = -2\sin\left(2x + \frac{\pi}{4}\right) \]

### Step 2: Analyze the sign of the derivative in the interval
Next, we need to analyze the sign of \( f'(x) \) in the interval \( \left[\frac{3\pi}{8}, \frac{5\pi}{8}\right] \).

1. Calculate the endpoints of the interval for \( 2x + \frac{\pi}{4} \):
- For \( x = \frac{3\pi}{8} \):
\[ 2x + \frac{\pi}{4} = 2 \cdot \frac{3\pi}{8} + \frac{\pi}{4} = \frac{3\pi}{4} + \frac{2\pi}{8} = \frac{3\pi}{4} + \frac{\pi}{4} = \pi \]
- For \( x = \frac{5\pi}{8} \):
\[ 2x + \frac{\pi}{4} = 2 \cdot \frac{5\pi}{8} + \frac{\pi}{4} = \frac{5\pi}{4} + \frac{2\pi}{8} = \frac{5\pi}{4} + \frac{\pi}{4} = \frac{6\pi}{4} = \frac{3\pi}{2} \]

So, \( 2x + \frac{\pi}{4} \) varies from \( \pi \) to \( \frac{3\pi}{2} \) as \( x \) goes from \( \frac{3\pi}{8} \) to \( \frac{5\pi}{8} \).

### Step 3: Determine the sign of \( \sin\left(2x + \frac{\pi}{4}\right) \)
- At \( 2x + \frac{\pi}{4} = \pi \), we have \( \sin(\pi) = 0 \).
- At \( 2x + \frac{\pi}{4} = \frac{3\pi}{2} \), we have \( \sin\left(\frac{3\pi}{2}\right) = -1 \).

Since \( \sin\left(2x + \frac{\pi}{4}\right) \) decreases from \( 0 \) to \( -1 \) in the interval \( \left[\pi, \frac{3\pi}{2}\right] \), it follows that:

\[ f'(x) = -2\sin\left(2x + \frac{\pi}{4}\right) \geq 0 \]

This means \( f'(x) \) is non-negative, indicating that \( f(x) \) is increasing in the interval \( \left[\frac{3\pi}{8}, \frac{5\pi}{8}\right] \).

### Conclusion
The function \( f(x) = \cos\left(2x + \frac{\pi}{4}\right) \) is increasing in the interval \( \left[\frac{3\pi}{8}, \frac{5\pi}{8}\right] \).

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