The points on the curve `f(x)=(x)/(1-x^(2))`, where the tangent to it has slope equal to unity, are `(x_(1), y_(1)), (x_(2), y_(2)) and (x_(3), y_(3))`. Then, `x_(1)+x_(2)+x_(3)` is equal to

Options

A. 0
B. `(sqrt3)/(2)`
C. `(sqrt3)/(4)`
D. `sqrt3`

Answer

To solve the problem, we need to find the points on the curve \( f(x) = \frac{x}{1 - x^2} \) where the slope of the tangent is equal to 1. We will follow these steps:

### Step 1: Differentiate the function
We start by finding the derivative of \( f(x) \) to determine the slope of the tangent line. We will use the quotient rule for differentiation.

\[ f'(x) = \frac{(1 - x^2)(1) - x(-2x)}{(1 - x^2)^2} \]

### Step 2: Simplify the derivative
Now we simplify the expression we obtained from the derivative.

\[ f'(x) = \frac{(1 - x^2) + 2x^2}{(1 - x^2)^2} = \frac{1 + x^2}{(1 - x^2)^2} \]

### Step 3: Set the derivative equal to 1
We need to find where the slope of the tangent is equal to 1, so we set the derivative equal to 1:

\[ \frac{1 + x^2}{(1 - x^2)^2} = 1 \]

### Step 4: Cross-multiply and simplify
Cross-multiplying gives us:

\[ 1 + x^2 = (1 - x^2)^2 \]

Expanding the right side:

\[ 1 + x^2 = 1 - 2x^2 + x^4 \]

### Step 5: Rearrange the equation
Rearranging the equation leads to:

\[ x^4 - 3x^2 = 0 \]

### Step 6: Factor the equation
We can factor out \( x^2 \):

\[ x^2(x^2 - 3) = 0 \]

### Step 7: Solve for x
Setting each factor to zero gives us:

1. \( x^2 = 0 \) → \( x = 0 \)
2. \( x^2 - 3 = 0 \) → \( x^2 = 3 \) → \( x = \sqrt{3}, -\sqrt{3} \)

Thus, the values of \( x \) are \( x_1 = 0, x_2 = \sqrt{3}, x_3 = -\sqrt{3} \).

### Step 8: Calculate \( x_1 + x_2 + x_3 \)
Now we can calculate the sum:

\[ x_1 + x_2 + x_3 = 0 + \sqrt{3} - \sqrt{3} = 0 \]

### Conclusion
Thus, the value of \( x_1 + x_2 + x_3 \) is \( \boxed{0} \).

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