For a function `f(x)=(2(x^(2)+1))/([x])` (where `[.]` denotes the greatest integer function), if `1lexlt4`. Then

Options

A. the range of f is `(4,(17)/(2))`
B. f is bijective function
C. the maximum value of `f(x)` is `(34)/(3)`
D. the minimum value of `f(x)` is 4

Answer

To solve the problem, we need to analyze the function \( f(x) = \frac{2(x^2 + 1)}{[x]} \) where \([x]\) denotes the greatest integer function, and \( 1 \leq x < 4 \). We will break down the solution into steps.

### Step 1: Identify the intervals for \( x \)
The given range for \( x \) is \( 1 \leq x < 4 \). We can break this interval into three parts based on the behavior of the greatest integer function:
1. \( 1 \leq x < 2 \)
2. \( 2 \leq x < 3 \)
3. \( 3 \leq x < 4 \)

### Step 2: Analyze the function in the first interval \( [1, 2) \)
In this interval, \([x] = 1\) for all \( x \) values. Therefore, the function simplifies to:
\[ f(x) = \frac{2(x^2 + 1)}{1} = 2(x^2 + 1) = 2x^2 + 2 \]
Now, we find the minimum and maximum values of \( f(x) \) in this interval:
- At \( x = 1 \):
\[ f(1) = 2(1^2) + 2 = 4 \]
- As \( x \) approaches \( 2 \) (but does not include it):
\[ f(1.9) = 2(1.9^2) + 2 \approx 2(3.61) + 2 = 7.22 \]
Thus, in the interval \( [1, 2) \), the range of \( f(x) \) is \( [4, 7.22) \).

### Step 3: Analyze the function in the second interval \( [2, 3) \)
In this interval, \([x] = 2\). Thus, the function becomes:
\[ f(x) = \frac{2(x^2 + 1)}{2} = x^2 + 1 \]
Now, we find the minimum and maximum values:
- At \( x = 2 \):
\[ f(2) = 2^2 + 1 = 5 \]
- As \( x \) approaches \( 3 \):
\[ f(2.9) = (2.9^2) + 1 \approx 8.41 \]
So, in the interval \( [2, 3) \), the range of \( f(x) \) is \( [5, 8.41) \).

### Step 4: Analyze the function in the third interval \( [3, 4) \)
In this interval, \([x] = 3\). Therefore, the function is:
\[ f(x) = \frac{2(x^2 + 1)}{3} = \frac{2x^2 + 2}{3} \]
Now, we find the minimum and maximum values:
- At \( x = 3 \):
\[ f(3) = \frac{2(3^2) + 2}{3} = \frac{20}{3} \approx 6.67 \]
- As \( x \) approaches \( 4 \):
\[ f(3.9) = \frac{2(3.9^2) + 2}{3} \approx \frac{30.42}{3} \approx 10.14 \]
Thus, in the interval \( [3, 4) \), the range of \( f(x) \) is \( [6.67, 10.14) \).

### Step 5: Combine the ranges from all intervals
Now, we have the ranges:
1. From \( [1, 2) \): \( [4, 7.22) \)
2. From \( [2, 3) \): \( [5, 8.41) \)
3. From \( [3, 4) \): \( [6.67, 10.14) \)

### Step 6: Determine the overall range
The overall range of \( f(x) \) is the union of these ranges:
- The minimum value is \( 4 \) (from the first interval).
- The maximum value is just below \( 10.14 \) (from the third interval).

Thus, the overall range of \( f(x) \) is:
\[ [4, 10.14) \]

### Conclusion
The function \( f(x) \) is not bijective because there are overlapping values in the ranges of different intervals. The minimum value is \( 4 \) and the maximum value approaches \( 10.14 \).