If `f(x) =sin^(4)x+cos^(4)x` increases, if

Options

A. `0ltxltpi//8`
B. `pi//40ltxlt3pi//8`
C. `3pi//8ltxlt5pi//8`
D. `5pi//8ltxlt3pi//4`

Explanation

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

### Step-by-Step Solution:

1. **Rewrite the Function**:
We start with the function:
\[ f(x) = \sin^4 x + \cos^4 x \]
We can simplify this using the identity \( a^2 + b^2 = (a + b)^2 - 2ab \):
\[ f(x) = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x \]
Since \( \sin^2 x + \cos^2 x = 1 \), we have:
\[ f(x) = 1 - 2\sin^2 x \cos^2 x \]

2. **Use the Double Angle Identity**:
We know that \( \sin 2x = 2\sin x \cos x \), so:
\[ \sin^2 x \cos^2 x = \frac{1}{4} \sin^2 2x \]
Thus, we can rewrite \( f(x) \):
\[ f(x) = 1 - \frac{1}{2} \sin^2 2x \]

3. **Differentiate the Function**:
Now, we differentiate \( f(x) \):
\[ f'(x) = 0 - \frac{1}{2} \cdot 2\sin 2x \cdot \cos 2x \cdot 2 \]
Simplifying this gives:
\[ f'(x) = -2\sin 2x \cos 2x \]
Using the double angle identity again, we have:
\[ f'(x) = -\sin 4x \]

4. **Determine Where the Function is Increasing**:
The function \( f(x) \) is increasing when \( f'(x) > 0 \):
\[ -\sin 4x > 0 \implies \sin 4x < 0 \]
The sine function is negative in the intervals:
\[ (n\pi, (n+1)\pi) \quad \text{for integers } n \]
Therefore, we need:
\[ 4x \in (n\pi, (n+1)\pi) \implies x \in \left(\frac{n\pi}{4}, \frac{(n+1)\pi}{4}\right) \]

5. **Identify Specific Intervals**:
For \( n = 0 \):
\[ x \in \left(0, \frac{\pi}{4}\right) \]
For \( n = 1 \):
\[ x \in \left(\frac{\pi}{4}, \frac{\pi}{2}\right) \]
For \( n = 2 \):
\[ x \in \left(\frac{\pi}{2}, \frac{3\pi}{4}\right) \]
Continuing this pattern, we can see that the function will be increasing in intervals of the form:
\[ x \in \left(\frac{n\pi}{4}, \frac{(n+1)\pi}{4}\right) \text{ for odd } n \]

### Summary of Intervals:
The function \( f(x) \) is increasing in the intervals:
- \( (0, \frac{\pi}{4}) \)
- \( (\frac{\pi}{2}, \frac{3\pi}{4}) \)
- and so on for other odd intervals.