If [x] denotes the greatest integer function then the extreme values of the function
`f(x)=[1+sinx]+[1+sin2x]+...+[1+sin nx], n in I^(+), x in (0,pi)` are

Options

A. (n-1)
B. n
C. (n+1)
D. (n+2)

Answer

To find the extreme values of the function

\[ f(x) = [1 + \sin x] + [1 + \sin 2x] + \ldots + [1 + \sin nx] \]

where \([x]\) denotes the greatest integer function, \(n \in \mathbb{I}^+\), and \(x \in (0, \pi)\), we can follow these steps:

### Step 1: Analyze the function components

The function can be rewritten as:

\[ f(x) = n + [\sin x] + [\sin 2x] + \ldots + [\sin nx] \]

Here, the term \(n\) comes from the fact that we are adding \(1\) a total of \(n\) times.

### Step 2: Determine the range of \(\sin kx\)

For \(x \in (0, \pi)\), the sine function \(\sin kx\) (for \(k = 1, 2, \ldots, n\)) will take values between \(0\) and \(1\).

- Specifically, \(\sin kx\) will reach its maximum value of \(1\) at \(x = \frac{\pi}{2k}\) for each \(k\) in the interval.

### Step 3: Apply the greatest integer function

Since \(\sin kx\) ranges from \(0\) to \(1\), we have:

\[ [\sin kx] = 0 \quad \text{for all } k \text{ in } (0, \pi) \]

This means that for all \(x\) in the interval \((0, \pi)\), the greatest integer function of \(\sin kx\) will yield \(0\).

### Step 4: Substitute back into the function

Thus, we can simplify \(f(x)\):

\[ f(x) = n + 0 + 0 + \ldots + 0 = n \]

### Step 5: Conclusion on extreme values

Since \(f(x) = n\) for all \(x \in (0, \pi)\), the extreme values of the function are constant and equal to \(n\).

### Final Answer

The extreme values of the function \(f(x)\) are:

\[ \text{Extreme values} = n \]