The function `f (x) = sin^4 x+cos^4 x` increases if

Options

A. ` 0 lt x lt pi //8`
B. ` pi //4 lt x lt 3 pi //8`
C. ` 3pi // 8 lt x lt 5 pi //8`
D. ` 5 pi // 8 lt x lt 3 pi //4`

Answer

To determine when the function \( f(x) = \sin^4 x + \cos^4 x \) is increasing, we need to analyze the first derivative of the function, \( f'(x) \).

### Step 1: Differentiate the function
We start by differentiating \( f(x) \):
\[ f'(x) = \frac{d}{dx}(\sin^4 x) + \frac{d}{dx}(\cos^4 x) \]
Using the chain rule:
\[ f'(x) = 4\sin^3 x \cdot \cos x - 4\cos^3 x \cdot \sin x \]

### Step 2: Factor out common terms
We can factor out common terms from \( f'(x) \):
\[ f'(x) = 4\sin x \cos x (\sin^2 x - \cos^2 x) \]
Using the double angle identity, \( \sin 2x = 2\sin x \cos x \), we rewrite:
\[ f'(x) = 2\sin 2x (\sin^2 x - \cos^2 x) \]

### Step 3: Set the derivative greater than or equal to zero
To find when \( f(x) \) is increasing, we need:
\[ f'(x) \geq 0 \implies 2\sin 2x (\sin^2 x - \cos^2 x) \geq 0 \]
This means we need to analyze the two factors \( \sin 2x \) and \( \sin^2 x - \cos^2 x \).

### Step 4: Analyze \( \sin 2x \)
The sine function is non-negative in the intervals:
\[ 2x \in [0, \pi] \quad \text{and} \quad 2x \in [2\pi, 3\pi] \quad \text{and so on.} \]
This translates to:
\[ x \in [0, \frac{\pi}{2}] \quad \text{and} \quad x \in [\pi, \frac{3\pi}{2}] \]

### Step 5: Analyze \( \sin^2 x - \cos^2 x \)
The expression \( \sin^2 x - \cos^2 x \) can be rewritten as \( -\cos 2x \). Thus, we need:
\[ -\cos 2x \geq 0 \implies \cos 2x \leq 0 \]
This occurs in the intervals:
\[ 2x \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \quad \text{and so on.} \]
This translates to:
\[ x \in \left(\frac{\pi}{4}, \frac{3\pi}{4}\right) \quad \text{and} \quad x \in \left(\frac{5\pi}{4}, \frac{7\pi}{4}\right) \]

### Step 6: Combine intervals
Now we combine the intervals where both conditions hold:
1. From \( \sin 2x \geq 0 \): \( x \in [0, \frac{\pi}{2}] \) or \( x \in [\pi, \frac{3\pi}{2}] \)
2. From \( \sin^2 x - \cos^2 x \geq 0 \): \( x \in \left(\frac{\pi}{4}, \frac{3\pi}{4}\right) \)

The intersection gives:
\[ x \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right] \]
Thus, the function \( f(x) \) is increasing in the interval \( \left[\frac{\pi}{4}, \frac{\pi}{2}\right] \).

### Final Result
The function \( f(x) = \sin^4 x + \cos^4 x \) increases in the interval:
\[ \left[\frac{\pi}{4}, \frac{\pi}{2}\right] \]