If `[x]^2- 5[x]+6= 0`, [x] denote the greatest integer function, then x ∈ [ 3 , 4 ] (ii)x ∈ ( 2 , 3 ](iii)x ∈ [ 2 , 4 )(iv)x ∈ [ 2 , 4 ]

Options

A. `x in[3,4]`
B. `x in (2,3]`
C. `x in [2,4)`
D. `x in [2,4]`

Explanation

To solve the equation \([x]^2 - 5[x] + 6 = 0\) where \([x]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Set up the equation
We start with the equation:
\[ [x]^2 - 5[x] + 6 = 0 \]

### Step 2: Factor the quadratic equation
We need to factor the quadratic expression. We look for two numbers that multiply to \(6\) (the constant term) and add to \(-5\) (the coefficient of the linear term). The numbers \(-2\) and \(-3\) satisfy this condition. Therefore, we can factor the equation as:
\[ ([x] - 2)([x] - 3) = 0 \]

### Step 3: Solve for \([x]\)
Setting each factor to zero gives us:
1. \([x] - 2 = 0 \implies [x] = 2\)
2. \([x] - 3 = 0 \implies [x] = 3\)

### Step 4: Determine the intervals for \(x\)
Now we need to find the intervals for \(x\) based on the values of \([x]\):
- If \([x] = 2\), then \(2 \leq x < 3\) (since \([x]\) is the greatest integer less than or equal to \(x\)).
- If \([x] = 3\), then \(3 \leq x < 4\).

### Step 5: Combine the intervals
Combining these intervals, we find:
- From \([x] = 2\): \(x \in [2, 3)\)
- From \([x] = 3\): \(x \in [3, 4)\)

Thus, the combined interval is:
\[ x \in [2, 4) \]

### Step 6: Identify the correct option
Now we can check the given options:
(i) \(x \in [3, 4]\)
(ii) \(x \in (2, 3]\)
(iii) \(x \in [2, 4)\)
(iv) \(x \in [2, 4]\)

The correct option based on our solution is:
**(iv) \(x \in [2, 4]\)**