If a function y=f(x) such that f'(x) is continuous function and satisfies
`(f(x))^(2)=k+int_(0)^(x) [{f(t)}^(2)+{f'(t)}^(2)]dt,k in R^(+) `, then

Options

A. f(x) is an increasing function for all ` x in R`
B. f(x) is a bounded function
C. f(x) is neither even nor odd function
D. If k=100, then f(0)=10.

Explanation

To solve the given problem, we start with the equation:

\[ (f(x))^2 = k + \int_0^x \left[ (f(t))^2 + (f'(t))^2 \right] dt \]

where \( k \) is a positive real number.

### Step 1: Differentiate both sides with respect to \( x \)

Using the Fundamental Theorem of Calculus and the chain rule, we differentiate both sides:

\[ \frac{d}{dx}((f(x))^2) = \frac{d}{dx}\left(k + \int_0^x \left[ (f(t))^2 + (f'(t))^2 \right] dt\right) \]

The left-hand side becomes:

\[ 2f(x)f'(x) \]

The right-hand side simplifies to:

\[ (f(x))^2 + (f'(x))^2 \]

Thus, we have:

\[ 2f(x)f'(x) = (f(x))^2 + (f'(x))^2 \]

### Step 2: Rearrange the equation

Rearranging gives:

\[ 2f(x)f'(x) - (f(x))^2 - (f'(x))^2 = 0 \]

### Step 3: Factor the equation

This can be factored as:

\[ (f'(x) - f(x))^2 = 0 \]

This implies:

\[ f'(x) - f(x) = 0 \]

### Step 4: Solve the differential equation

The differential equation \( f'(x) = f(x) \) can be solved by separating variables:

\[ \frac{df}{f} = dx \]

Integrating both sides gives:

\[ \ln |f(x)| = x + C \]

Exponentiating both sides results in:

\[ f(x) = Ce^x \]

where \( C = e^C \) is a constant.

### Step 5: Determine the value of \( C \)

To find \( C \), we substitute \( x = 0 \) into the original equation:

\[ (f(0))^2 = k + \int_0^0 \left[ (f(t))^2 + (f'(t))^2 \right] dt \]

This simplifies to:

\[ (f(0))^2 = k \]

Since \( f(0) = Ce^0 = C \), we have:

\[ C^2 = k \implies C = \pm \sqrt{k} \]

### Step 6: Analyze the function

Thus, we have:

\[ f(x) = \pm \sqrt{k} e^x \]

### Conclusion

Now we analyze the properties of \( f(x) \):

1. **Increasing Function**: Since \( e^x > 0 \) for all \( x \), \( f(x) \) is an increasing function for all \( x \in \mathbb{R} \).
2. **Bounded Function**: \( f(x) \) is not bounded as \( e^x \) grows without bound.
3. **Even or Odd Function**: \( f(x) \) is neither even nor odd since \( f(-x) \neq f(x) \) and \( f(-x) \neq -f(x) \).
4. **Specific Value of \( k \)**: If \( k = 100 \), then \( C = 10 \) or \( C = -10 \).