Let `f(x)=(x^(2)+1)/([x]),1 lt x le 3.9.[.]` denotes the greatest integer function. Then

A. (a)f (x) is increasing function
B. f (x) is decreasing function
C. The greatest value of f (x) is `(1)/(3)xx16.21`
D. The least value of f (x) is 2

Options

A. (a)f (x) is increasing function
B. f (x) is decreasing function
C. The greatest value of f (x) is `(1)/(3)xx16.21`
D. The least value of f (x) is 2

Explanation

To solve the problem, we need to analyze the function \( f(x) = \frac{x^2 + 1}{[x]} \) where \( [x] \) is the greatest integer function, and the domain of \( x \) is \( 1 < x \leq 3.9 \).

### Step 1: Determine the intervals based on the greatest integer function
The greatest integer function \( [x] \) will take on different integer values depending on the range of \( x \):
- For \( 1 < x \leq 2 \), \( [x] = 1 \)
- For \( 2 < x \leq 3 \), \( [x] = 2 \)
- For \( 3 < x \leq 3.9 \), \( [x] = 3 \)

### Step 2: Define the function in each interval
Now we can express \( f(x) \) in each of these intervals:
1. For \( 1 < x \leq 2 \):
\[ f(x) = \frac{x^2 + 1}{1} = x^2 + 1 \]
2. For \( 2 < x \leq 3 \):
\[ f(x) = \frac{x^2 + 1}{2} \]
3. For \( 3 < x \leq 3.9 \):
\[ f(x) = \frac{x^2 + 1}{3} \]

### Step 3: Analyze each interval
#### Interval 1: \( 1 < x \leq 2 \)
- The function \( f(x) = x^2 + 1 \) is a quadratic function which is increasing in this interval.
- Calculate \( f(1) \) and \( f(2) \):
- \( f(1) = 1^2 + 1 = 2 \) (not included)
- \( f(2) = 2^2 + 1 = 5 \)

#### Interval 2: \( 2 < x \leq 3 \)
- The function \( f(x) = \frac{x^2 + 1}{2} \) is also increasing since \( x^2 \) is increasing.
- Calculate \( f(2) \) and \( f(3) \):
- \( f(2) = \frac{2^2 + 1}{2} = \frac{5}{2} = 2.5 \)
- \( f(3) = \frac{3^2 + 1}{2} = \frac{10}{2} = 5 \)

#### Interval 3: \( 3 < x \leq 3.9 \)
- The function \( f(x) = \frac{x^2 + 1}{3} \) is increasing as well.
- Calculate \( f(3) \) and \( f(3.9) \):
- \( f(3) = \frac{10}{3} \approx 3.33 \)
- \( f(3.9) = \frac{(3.9)^2 + 1}{3} = \frac{16.21}{3} \approx 5.40 \)

### Step 4: Determine the greatest and least values
- The least value of \( f(x) \) occurs at \( x \) approaching \( 1 \), which is \( 2 \).
- The greatest value occurs at \( x = 3.9 \), which is \( \frac{16.21}{3} \).

### Conclusion
- The greatest value of \( f(x) \) is \( \frac{16.21}{3} \).
- The least value of \( f(x) \) is \( 2 \).
- Therefore, the correct options are:
- C. The greatest value of \( f(x) \) is \( \frac{16.21}{3} \).
- D. The least value of \( f(x) \) is \( 2 \).