Consider f, g and h be three real valued function defined on R. Let `f(x)=sin3x+cosx,g(x)=cos3x+sinx` and `h(x)=f^(2)(x)+g^(2)(x).` Then,
The length of a longest interval in which the function g=h(x) is increasing, is

Options

A. `(pi)/(8)`
B. `(pi)/(4)`
C. `(pi)/(6)`
D. `(pi)/(2)`

Answer

To solve the problem, we need to analyze the functions \( f(x) \), \( g(x) \), and \( h(x) \) given as follows:

1. \( f(x) = \sin(3x) + \cos(x) \)
2. \( g(x) = \cos(3x) + \sin(x) \)
3. \( h(x) = f^2(x) + g^2(x) \)

We want to find the longest interval in which the function \( g(x) = h(x) \) is increasing.

### Step 1: Calculate \( h(x) \)

First, we need to compute \( h(x) = f^2(x) + g^2(x) \).

\[ h(x) = (\sin(3x) + \cos(x))^2 + (\cos(3x) + \sin(x))^2 \]

Expanding both squares:

\[ h(x) = \sin^2(3x) + 2\sin(3x)\cos(x) + \cos^2(x) + \cos^2(3x) + 2\cos(3x)\sin(x) + \sin^2(x) \]

Using the Pythagorean identity \( \sin^2(a) + \cos^2(a) = 1 \):

\[ h(x) = 1 + 1 + 2(\sin(3x)\cos(x) + \cos(3x)\sin(x)) \]

This simplifies to:

\[ h(x) = 2 + 2\sin(3x + x) = 2 + 2\sin(4x) \]

Thus, we have:

\[ h(x) = 2 + 2\sin(4x) \]

### Step 2: Find the Derivative of \( h(x) \)

Next, we find the derivative \( h'(x) \) to determine when \( h(x) \) is increasing.

\[ h'(x) = \frac{d}{dx}(2 + 2\sin(4x)) = 2 \cdot 4\cos(4x) = 8\cos(4x) \]

### Step 3: Determine When \( h'(x) > 0 \)

The function \( h(x) \) is increasing when \( h'(x) > 0 \):

\[ 8\cos(4x) > 0 \]

This implies:

\[ \cos(4x) > 0 \]

The cosine function is positive in the intervals:

\[ 4x \in (2k\pi, (2k+1)\pi) \quad \text{for } k \in \mathbb{Z} \]

Dividing by 4 gives:

\[ x \in \left(\frac{k\pi}{2}, \frac{(2k+1)\pi}{4}\right) \]

### Step 4: Determine the Length of the Interval

The length of one interval \( \left(\frac{k\pi}{2}, \frac{(2k+1)\pi}{4}\right) \) can be calculated as follows:

\[ \text{Length} = \frac{(2k+1)\pi}{4} - \frac{k\pi}{2} \]

This simplifies to:

\[ \text{Length} = \frac{(2k+1)\pi}{4} - \frac{2k\pi}{4} = \frac{\pi}{4} \]

### Conclusion

The length of the longest interval in which the function \( g(x) = h(x) \) is increasing is:

\[ \boxed{\frac{\pi}{4}} \]