The number of solution of the equation
`16(x^(2)+1)+pi^(2)=|tanx|+8pix` is equal to

Options

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

Answer

To solve the equation \( 16(x^2 + 1) + \pi^2 = | \tan x | + 8\pi x \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 16(x^2 + 1) + \pi^2 = | \tan x | + 8\pi x \]
This can be rewritten as:
\[ 16x^2 + 16 + \pi^2 - 8\pi x = | \tan x | \]

### Step 2: Define a new function
Let us define a new function:
\[ f(x) = 16x^2 - 8\pi x + 16 + \pi^2 \]
We need to analyze the function \( f(x) \) and compare it with \( | \tan x | \).

### Step 3: Analyze the function \( f(x) \)
The function \( f(x) \) is a quadratic function in the form \( ax^2 + bx + c \), where:
- \( a = 16 \) (which is positive, indicating the parabola opens upwards),
- \( b = -8\pi \),
- \( c = 16 + \pi^2 \).

### Step 4: Calculate the discriminant
To find the number of solutions, we need to calculate the discriminant \( D \) of the quadratic equation:
\[ D = b^2 - 4ac \]
Substituting the values:
\[ D = (-8\pi)^2 - 4 \cdot 16 \cdot (16 + \pi^2) \]
Calculating \( D \):
\[ D = 64\pi^2 - 64(16 + \pi^2) = 64\pi^2 - 1024 - 64\pi^2 = -1024 \]

### Step 5: Determine the nature of the roots
Since the discriminant \( D \) is negative (\( D < 0 \)), the quadratic function \( f(x) \) does not intersect the x-axis. This means \( f(x) \) is always positive for all \( x \).

### Step 6: Compare with \( | \tan x | \)
The function \( | \tan x | \) is periodic and can take any non-negative value, including values that can be less than or equal to \( f(x) \). However, since \( f(x) \) is always positive, there will be points where \( | \tan x | \) equals \( f(x) \).

### Step 7: Conclusion on the number of solutions
Since \( | \tan x | \) can take values from \( 0 \) to \( \infty \) and \( f(x) \) is always positive, we conclude that the number of solutions to the equation \( 16(x^2 + 1) + \pi^2 = | \tan x | + 8\pi x \) is infinite.

### Final Answer:
The number of solutions of the equation is infinite.

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