Statement I The function `f(x) = int_(0)^(x) sqrt(1+t^(2) dt )` is an odd function and STATEMENT 2 :`g(x)=f'(x)` is an even function , then `f(x)` is an odd function.

Options

A. Statement I is true, Statement II is also true, Statement II is the correct explanation of Statement 1.
B. Statement I is true, Statement II is also true , Statement II is not the correct explanation of Statement II.
C. Statement I is true, Statement II is false
D. Statement I is false , Statement II is true

Answer

To solve the problem, we need to analyze the statements given regarding the function \( f(x) \) and its derivative \( g(x) = f'(x) \).

### Step 1: Understanding the Function \( f(x) \)

The function is defined as:
\[ f(x) = \int_{0}^{x} \sqrt{1 + t^2} \, dt \]
We need to determine if \( f(x) \) is an odd function. A function \( f(x) \) is odd if:
\[ f(-x) = -f(x) \]

### Step 2: Finding \( f(-x) \)

To find \( f(-x) \), we substitute \(-x\) into the integral:
\[ f(-x) = \int_{0}^{-x} \sqrt{1 + t^2} \, dt \]
Using the property of definite integrals, we can change the limits:
\[ f(-x) = -\int_{-x}^{0} \sqrt{1 + t^2} \, dt \]
This can be rewritten as:
\[ f(-x) = -\int_{0}^{x} \sqrt{1 + (-u)^2} \, (-du) = -\int_{0}^{x} \sqrt{1 + u^2} \, du \]
where we made a substitution \( t = -u \).

### Step 3: Simplifying \( f(-x) \)

Thus, we have:
\[ f(-x) = -\int_{0}^{x} \sqrt{1 + u^2} \, du = -f(x) \]
This confirms that \( f(x) \) is indeed an odd function.

### Step 4: Analyzing the Derivative \( g(x) = f'(x) \)

Next, we need to find \( g(x) \):
\[ g(x) = f'(x) = \sqrt{1 + x^2} \]

### Step 5: Checking if \( g(x) \) is Even

To check if \( g(x) \) is an even function, we evaluate \( g(-x) \):
\[ g(-x) = f'(-x) = \sqrt{1 + (-x)^2} = \sqrt{1 + x^2} = g(x) \]
Since \( g(-x) = g(x) \), we conclude that \( g(x) \) is an even function.

### Conclusion

- **Statement 1**: \( f(x) \) is an odd function. **True**.
- **Statement 2**: \( g(x) = f'(x) \) is an even function. **True**.

Thus, both statements are true, and Statement 2 correctly explains Statement 1.

### Final Answer

Both statements are true, and Statement 2 is a correct explanation of Statement 1.

---