Let f(x) = `x - x^(2) and g(x) = {x}, AA x in R` where denotes fractional part function.
Statement I f(g(x)) will be continuous, `AA x in R`.
Statement II `f(0) = f(1) and g(x)` is periodic with period 1.

Options

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

Answer

To solve the problem, we need to analyze the two statements given about the functions \( f(x) \) and \( g(x) \).

1. **Define the Functions:**
- \( f(x) = x - x^2 \)
- \( g(x) = \{x\} \) (the fractional part of \( x \))

2. **Evaluate Statement II:**
- We need to check if \( f(0) = f(1) \) and if \( g(x) \) is periodic with period 1.
- Calculate \( f(0) \):
\[ f(0) = 0 - 0^2 = 0 \]
- Calculate \( f(1) \):
\[ f(1) = 1 - 1^2 = 1 - 1 = 0 \]
- Thus, \( f(0) = f(1) \) is true.
- Now, check if \( g(x) \) is periodic with period 1. The fractional part function \( g(x) = \{x\} \) is defined as:
\[ g(x) = x - \lfloor x \rfloor \]
This function repeats its values for every integer increment, confirming that it is periodic with period 1.

**Conclusion for Statement II:**
- Both parts of Statement II are true: \( f(0) = f(1) \) and \( g(x) \) is periodic with period 1.

3. **Evaluate Statement I:**
- We need to check if \( f(g(x)) \) is continuous for all \( x \in \mathbb{R} \).
- Substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(\{x\}) = \{x\} - \{x\}^2 \]
- The function \( f(g(x)) = \{x\} - \{x\}^2 \) is a composition of continuous functions:
- The fractional part function \( g(x) \) is continuous everywhere.
- The function \( f(x) = x - x^2 \) is a polynomial and is continuous everywhere.

- Since the composition of continuous functions is continuous, \( f(g(x)) \) is continuous for all \( x \in \mathbb{R} \).

**Conclusion for Statement I:**
- Statement I is true.

4. **Final Conclusion:**
- Both statements are true, and Statement II provides a correct explanation for Statement I.

Thus, the answer is that both statements are true.