let `f (x) = sin ^(-1) ((2g (x))/(1+g (x)^(2))),` then which are correct ?
(i) f (x) is decreasing if `g (x)` is increasig and `|g (x) gt 1`
(ii) `f (x)` is an increasing function if `g (x)` is increasing and `|g (x) |le 1`
(iii) f (x) is decreasing function if ` f(x)` is decreasing and `|g (x) | gt 1`

Options

A. (i) and (iii)
B. (i) and (ii)
C. (i) (ii) and (iii)
D. (iii)

Answer

To solve the problem, we need to analyze the function \( f(x) = \sin^{-1} \left( \frac{2g(x)}{1 + g(x)^2} \right) \) and determine the behavior of \( f(x) \) based on the properties of \( g(x) \).

### Step 1: Understanding the Function
The function \( f(x) \) is defined as:
\[ f(x) = \sin^{-1} \left( \frac{2g(x)}{1 + g(x)^2} \right) \]
This expression is valid for \( |g(x)| \leq 1 \) because the range of \( \sin^{-1} \) is limited to \([-1, 1]\).

### Step 2: Analyzing the Derivative
To determine whether \( f(x) \) is increasing or decreasing, we need to compute the derivative \( f'(x) \).

Using the chain rule:
\[ f'(x) = \frac{d}{dx} \left( \sin^{-1}(u) \right) \cdot \frac{du}{dx} \]
where \( u = \frac{2g(x)}{1 + g(x)^2} \).

The derivative of \( \sin^{-1}(u) \) is:
\[ \frac{1}{\sqrt{1 - u^2}} \]

Now, we need to find \( \frac{du}{dx} \):
\[ u = \frac{2g(x)}{1 + g(x)^2} \]
Using the quotient rule:
\[ \frac{du}{dx} = \frac{(1 + g(x)^2)(2g'(x)) - 2g(x)(2g(x)g'(x))}{(1 + g(x)^2)^2} \]
Simplifying this gives:
\[ \frac{du}{dx} = \frac{2g'(x)(1 + g(x)^2 - 2g(x)^2)}{(1 + g(x)^2)^2} = \frac{2g'(x)(1 - g(x)^2)}{(1 + g(x)^2)^2} \]

Thus, we have:
\[ f'(x) = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{2g'(x)(1 - g(x)^2)}{(1 + g(x)^2)^2} \]

### Step 3: Analyzing the Sign of \( f'(x) \)
1. **For \( |g(x)| < 1 \)**:
- If \( g(x) \) is increasing (\( g'(x) > 0 \)), then \( 1 - g(x)^2 > 0 \) implies \( f'(x) > 0 \). Therefore, \( f(x) \) is increasing.

2. **For \( |g(x)| > 1 \)**:
- If \( g(x) \) is increasing (\( g'(x) > 0 \)), then \( 1 - g(x)^2 < 0 \) implies \( f'(x) < 0 \). Therefore, \( f(x) \) is decreasing.

### Step 4: Evaluating the Statements
Now we can evaluate the statements given in the problem:

1. **Statement (i)**: \( f(x) \) is decreasing if \( g(x) \) is increasing and \( |g(x)| > 1 \).
- This is **correct** based on our analysis.

2. **Statement (ii)**: \( f(x) \) is increasing if \( g(x) \) is increasing and \( |g(x)| \leq 1 \).
- This is also **correct** based on our analysis.

3. **Statement (iii)**: \( f(x) \) is decreasing if \( g(x) \) is decreasing and \( |g(x)| > 1 \).
- This is **incorrect** because if \( g(x) \) is decreasing (\( g'(x) < 0 \)), then \( f'(x) \) could be positive or negative depending on the value of \( g(x) \).

### Conclusion
The correct statements are:
- (i) and (ii) are correct.
- (iii) is incorrect.