STATEMENT-1: Let f(x) and g(x) be two decreasing function then f(g(x)) must be an increasing function .
and
STATEMENT-2 : `f(g(2)) gt f(g(1))` , where f and g are two decreasing function .

Options

A. Statement-1 is True , Statement-2 is True , Statement-2 is a correct explanation for Statement-1 .
B. Statement-1 is True , Statement-2 is True , Statement-2 is NOT a correct explanation for Statement-1 .
C. Statement-1 is True , Statement-2 is False
D. Statement-1 is False , Statement-2 is True

Answer

To solve the problem, we need to analyze the two statements given:

**Statement 1**: Let \( f(x) \) and \( g(x) \) be two decreasing functions. Then \( f(g(x)) \) must be an increasing function.

**Statement 2**: \( f(g(2)) > f(g(1)) \), where \( f \) and \( g \) are two decreasing functions.

### Step-by-Step Solution:

1. **Understanding Decreasing Functions**:
- A function \( f(x) \) is decreasing if for any two values \( x_1 \) and \( x_2 \) such that \( x_1 < x_2 \), we have \( f(x_1) \geq f(x_2) \).
- Similarly, \( g(x) \) is also a decreasing function.

2. **Analyzing Statement 1**:
- Since \( g(x) \) is decreasing, if \( x_2 > x_1 \), then \( g(x_2) < g(x_1) \).
- Now, since \( f(x) \) is also decreasing, applying \( f \) to the outputs of \( g \):
- If \( g(x_2) < g(x_1) \), then \( f(g(x_2)) > f(g(x_1)) \).
- This shows that \( f(g(x)) \) is increasing because as \( x \) increases, \( f(g(x)) \) increases.

3. **Conclusion for Statement 1**:
- Therefore, Statement 1 is **True**: \( f(g(x)) \) is indeed an increasing function.

4. **Analyzing Statement 2**:
- We need to check if \( f(g(2)) > f(g(1)) \).
- Since \( g(x) \) is decreasing, it follows that \( g(2) < g(1) \).
- Now applying \( f \) (which is also decreasing), since \( g(2) < g(1) \), we have:
- \( f(g(2)) > f(g(1)) \).
- This confirms that Statement 2 is also **True**.

5. **Final Conclusion**:
- Both statements are true, but Statement 2 does not serve as a correct expression of Statement 1. Thus, we conclude that:
- Statement 1 is true.
- Statement 2 is true.
- However, Statement 2 is not a correct expression for Statement 1.