Consider f,g and h be three real valued functions defined on R. Let `f(x)={:{(-1", "xlt0),(0", "x=0","g(x)(1-x^(2))andh(x) "be such that"),(1", "xgto):}` h''(x)=6x-4. Also, h(x) has local minimum value 5 at x=1
Range of function `sin^(-1)sqrt((fog(x)))` is

Options

A. `(0,pi//2)`
B. `{0,pi//2}`
C. `{-[pi//2,0,pi//2}`
D. `{pi//2}`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Functions
We have three functions defined:
- \( f(x) \) is a piecewise function:
\[ f(x) = \begin{cases} -1 & \text{if } x < 0 \\ 0 & \text{if } x = 0 \\ 1 & \text{if } x > 0 \end{cases} \]
- \( g(x) = 1 - x^2 \)
- \( h(x) \) is defined such that \( h''(x) = 6x - 4 \) and has a local minimum value of 5 at \( x = 1 \).

### Step 2: Analyze \( g(x) \)
The function \( g(x) = 1 - x^2 \) is a downward-opening parabola. The maximum value occurs at \( x = 0 \):
- \( g(0) = 1 \)
- As \( x \) approaches \( \pm \infty \), \( g(x) \) approaches \(-\infty\).

The range of \( g(x) \) is \( (-\infty, 1] \).

### Step 3: Determine the Values of \( f(g(x)) \)
Now, we need to find \( f(g(x)) \):
- For \( g(x) \leq 0 \) (which occurs when \( |x| \geq 1 \)), \( f(g(x)) = -1 \).
- For \( g(x) = 1 \) (which occurs at \( x = 0 \)), \( f(g(x)) = 1 \).
- For \( 0 < g(x) < 1 \) (which occurs when \( -1 < x < 1 \)), \( f(g(x)) = 1 \).

Thus, we can summarize:
- \( f(g(x)) = -1 \) for \( |x| \geq 1 \)
- \( f(g(x)) = 1 \) for \( -1 < x < 1 \)
- \( f(g(x)) = 0 \) at \( x = 0 \)

### Step 4: Find the Range of \( \sin^{-1}(\sqrt{f(g(x))}) \)
Next, we need to find the range of \( \sin^{-1}(\sqrt{f(g(x))}) \):
- The possible values of \( f(g(x)) \) are \(-1\), \(0\), and \(1\).
- The square root function is only defined for non-negative values, so we only consider \( f(g(x)) = 0 \) and \( f(g(x)) = 1 \):
- \( \sqrt{0} = 0 \)
- \( \sqrt{1} = 1 \)

### Step 5: Calculate \( \sin^{-1} \)
Now we find:
- \( \sin^{-1}(0) = 0 \)
- \( \sin^{-1}(1) = \frac{\pi}{2} \)

### Final Step: Conclusion
The range of the function \( \sin^{-1}(\sqrt{f(g(x))}) \) is:
\[ \text{Range} = \{ 0, \frac{\pi}{2} \} \]