`(1)/(x^(3)), (1)/(x^(2)), (1)/(x), x ,x^(2),x^(3)`
If `-1 lt x lt 0`, what is the median of the six numbers in the list above ?

Options

A. `(1)/(x)`
B. `x^(2)`
C. `(x^(2)(x+1))/(2)`
D. `(x(x^(2)+1))/(2)`

Answer

To find the median of the six numbers given the condition \(-1 < x < 0\), we will follow these steps:

### Step 1: List the Numbers
The numbers given are:
\[ \frac{1}{x^3}, \frac{1}{x^2}, \frac{1}{x}, x, x^2, x^3 \]

### Step 2: Determine the Order of the Numbers
Since \(x\) is negative (between -1 and 0), we need to analyze the order of these numbers. We can evaluate the terms based on the properties of negative numbers:

1. **For \(x^3\)**: Since \(x\) is negative, \(x^3\) will also be negative and will be less than \(x^2\) (which is positive).
2. **For \(x^2\)**: This is positive and greater than \(x\) and \(x^3\).
3. **For \(\frac{1}{x}\)**: This will be negative and greater than \(x^3\) but less than \(x^2\).
4. **For \(\frac{1}{x^2}\)**: This will be positive and greater than both \(\frac{1}{x}\) and \(x^2\).
5. **For \(\frac{1}{x^3}\)**: This will be positive and greater than \(\frac{1}{x^2}\).

Thus, the order of the numbers when \( -1 < x < 0 \) is:
\[ \frac{1}{x^3}, \frac{1}{x^2}, \frac{1}{x}, x, x^2, x^3 \]

### Step 3: Identify the Median
The median is the average of the two middle numbers in an ordered list. Since we have 6 numbers, the median will be the average of the 3rd and 4th numbers.

From our ordered list:
- The 3rd term is \(x\).
- The 4th term is \(x^3\).

### Step 4: Calculate the Median
The median is calculated as follows:
\[ \text{Median} = \frac{x + x^3}{2} \]

### Step 5: Final Expression
We can factor out \(x\) from the expression:
\[ \text{Median} = \frac{x(1 + x^2)}{2} \]

### Conclusion
Thus, the median of the six numbers in the list is:
\[ \frac{x(1 + x^2)}{2} \]