Let `h(x)=f(x)=f_(x)-g_(x)`, where `f_(x)=sin^(4)pix` and `g(x)=In x`. Let `x_(0),x_(1),x_(2) , ....,x_(n+1_` be the roots of `f_(x)=g_(x)` in increasing order.
In the above question, the value of n is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the value of \( n \) in the equation \( h(x) = f(x) - g(x) \) where \( f(x) = \sin^4(4\pi x) \) and \( g(x) = \ln x \), we need to determine the number of intersections (roots) between the two functions \( f(x) \) and \( g(x) \) in the specified range.

### Step-by-Step Solution:

1. **Identify the Functions**:
- We have \( f(x) = \sin^4(4\pi x) \) and \( g(x) = \ln x \).

2. **Determine the Range of \( f(x) \)**:
- The function \( \sin^4(4\pi x) \) oscillates between 0 and 1, as the sine function oscillates between -1 and 1. Therefore, \( f(x) \) will also oscillate between 0 and 1.

3. **Determine the Range of \( g(x) \)**:
- The function \( g(x) = \ln x \) is defined for \( x > 0 \) and approaches \( -\infty \) as \( x \) approaches 0. It increases without bound as \( x \) increases.

4. **Find Points of Intersection**:
- We need to find the points where \( \sin^4(4\pi x) = \ln x \).

5. **Graphical Representation**:
- The graph of \( \sin^4(4\pi x) \) will have periodic peaks at \( x = \frac{k}{4} \) for \( k = 0, 1, 2, \ldots \) (where \( k \) is an integer), with each peak reaching a maximum of 1.
- The graph of \( \ln x \) will intersect these peaks.

6. **Determine the Number of Intersections**:
- The function \( \sin^4(4\pi x) \) completes one full cycle (from 0 to 1 back to 0) every \( \frac{1}{4} \) units. Therefore, in the interval \( (0, 2) \), it completes \( 8 \) cycles (since \( 2 \div \frac{1}{4} = 8 \)).
- Each cycle has two intersections with \( \ln x \) (one on the way up and one on the way down), leading to \( 8 \times 2 = 16 \) intersections.

7. **Final Count of Roots**:
- The roots are denoted as \( x_0, x_1, x_2, \ldots, x_{n+1} \). Therefore, if there are \( 16 \) intersections, then \( n + 1 = 16 \), which gives us \( n = 15 \).

### Conclusion:
The value of \( n \) is \( 15 \).