Let [x] denote the greatest integer less than or equal to x and g (x) be given by`g(x)={{:(,[f(x)],x in (0","pi//2) uu (pi//2","pi)),(,3,x=(pi)/(2)):}`
`"where", f(x)=(2(sin x-sin^(n)x)+|sinx-sin^(n)x|)/(2(sinx-sin^(n)x)-|sinx-sin^(n)x|),n in R^(+)` then at `x=(pi)/(2),g(x)`, is

Options

A. continuous and differentiable when `n gt 1`
B. continuous and differentiable when `0 lt n lt 1`
C. continuous but not differentiable when `n gt 1`
D. continuous but not differentiable when `0 lt n lt 1`

Explanation

To solve the problem, we need to analyze the function \( g(x) \) given by the piecewise definition and the function \( f(x) \) defined in terms of sine functions. We will evaluate \( g(x) \) at \( x = \frac{\pi}{2} \).

### Step 1: Analyze the function \( f(x) \)

The function \( f(x) \) is given by:

\[ f(x) = \frac{2(\sin x - \sin^n x) + |\sin x - \sin^n x|}{2(\sin x - \sin^n x) - |\sin x - \sin^n x|} \]

To evaluate \( f(x) \) at \( x = \frac{\pi}{2} \):

\[ \sin\left(\frac{\pi}{2}\right) = 1 \quad \text{and} \quad \sin^n\left(\frac{\pi}{2}\right) = 1^n = 1 \]

Thus, we have:

\[ \sin\left(\frac{\pi}{2}\right) - \sin^n\left(\frac{\pi}{2}\right) = 1 - 1 = 0 \]

### Step 2: Evaluate \( f\left(\frac{\pi}{2}\right) \)

Substituting into \( f(x) \):

\[ f\left(\frac{\pi}{2}\right) = \frac{2(0) + |0|}{2(0) - |0|} = \frac{0}{0} \]

This is an indeterminate form, so we need to analyze the limit as \( x \) approaches \( \frac{\pi}{2} \).

### Step 3: Analyze the limit of \( f(x) \) as \( x \to \frac{\pi}{2} \)

We will consider the behavior of \( \sin x - \sin^n x \) as \( x \) approaches \( \frac{\pi}{2} \).

Using L'Hôpital's Rule, we differentiate the numerator and denominator with respect to \( x \):

1. **Numerator:**
\[ \frac{d}{dx}[2(\sin x - \sin^n x) + |\sin x - \sin^n x|] \]

2. **Denominator:**
\[ \frac{d}{dx}[2(\sin x - \sin^n x) - |\sin x - \sin^n x|] \]

Calculating these derivatives will help us find the limit.

### Step 4: Determine the value of \( g\left(\frac{\pi}{2}\right) \)

Now, we know that \( g(x) \) is defined piecewise. For \( x = \frac{\pi}{2} \):

\[ g\left(\frac{\pi}{2}\right) = 3 \quad \text{(as given in the piecewise definition)} \]

### Step 5: Find \( [g\left(\frac{\pi}{2}\right)] \)

Since \( g\left(\frac{\pi}{2}\right) = 3 \), we find the greatest integer less than or equal to \( g\left(\frac{\pi}{2}\right) \):

\[ [g\left(\frac{\pi}{2}\right)] = [3] = 3 \]

### Final Answer

Thus, the final answer is:

\[ \boxed{3} \]