The complete solution set of the inequality` [cot^(-1)x]^2-6[cot^(-1)x]+9leq0` where [ ] denotes greatest integer function, is

Options

A. `(-oo,cot3]`
B. `[cot3,cot2]`
C. `[cot3,oo)`
D. `(-oo,cot2]`

Explanation

To solve the inequality \([cot^{-1}x]^2 - 6[cot^{-1}x] + 9 \leq 0\), where \([ ]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Substitute \( t \)
Let \( t = [cot^{-1}x] \). The inequality then becomes:
\[ t^2 - 6t + 9 \leq 0 \]

### Step 2: Factor the quadratic
We can factor the quadratic expression:
\[ t^2 - 6t + 9 = (t - 3)^2 \]
Thus, the inequality can be rewritten as:
\[ (t - 3)^2 \leq 0 \]

### Step 3: Analyze the inequality
The expression \((t - 3)^2\) is a perfect square and is always non-negative. Therefore, the only time it equals zero is when:
\[ t - 3 = 0 \implies t = 3 \]

### Step 4: Relate back to \( cot^{-1}x \)
Since \( t = [cot^{-1}x] \), we have:
\[ [cot^{-1}x] = 3 \]
This means:
\[ 3 \leq cot^{-1}x < 4 \]

### Step 5: Find the range of \( x \)
To find the values of \( x \) that satisfy this inequality, we take the cotangent of the bounds:
\[ cot(3) < x \leq cot(4) \]

### Step 6: Final solution
Thus, the complete solution set for the inequality is:
\[ x \in (cot(4), cot(3)] \]