The differentiable function `y= f(x)` has a property that the chord joining any two points `A (x _(1), f (x_(1)) and B (x_(2), f(x _(2)))` always intersects y-axis at `(0,2 x _(1) x _(2)).` Given that `f (1) =-1.` then:
The largest interval in which `y =f (x)` is monotonically increasing, is : (a) `(-oo,(1)/(2)]` (b) `[(-1)/(2),oo)` (c) `(-oo, (1)/(4)]` (d) ` [(-1)/(4), oo)`

Options

A. `(-oo,(1)/(2)]`
B. `[(-1)/(2),oo)`
C. `(-oo, (1)/(4)]`
D. ` [(-1)/(4), oo)`

Answer

To solve the problem, we need to find the largest interval in which the function \( y = f(x) \) is monotonically increasing. We start with the given property of the function and the point \( f(1) = -1 \).

### Step-by-step Solution:

1. **Understanding the Chord Property**:
The chord joining points \( A(x_1, f(x_1)) \) and \( B(x_2, f(x_2)) \) intersects the y-axis at the point \( (0, 2x_1x_2) \). The equation of the chord can be expressed using the slope formula:
\[ y - f(x_1) = m(x - x_1) \]
where \( m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \).

2. **Setting the Intersection Point**:
Since the chord intersects the y-axis at \( (0, 2x_1x_2) \), we substitute \( x = 0 \):
\[ 2x_1x_2 - f(x_1) = m(-x_1) \]
Rearranging gives:
\[ 2x_1x_2 = -mx_1 + f(x_1) \]

3. **Substituting for the Slope**:
We can express \( m \) in terms of \( f(x_1) \) and \( f(x_2) \):
\[ m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]
Substituting this back into our equation, we can derive a relationship involving \( f(x_1) \) and \( f(x_2) \).

4. **Finding the Function**:
After manipulating the equation, we find that:
\[ f(x) = x - 2x^2 \]
This is derived from the property of the chord and the given point \( f(1) = -1 \).

5. **Differentiating the Function**:
To find where the function is increasing, we differentiate:
\[ f'(x) = 1 - 4x \]

6. **Setting the Derivative Greater than Zero**:
For \( f(x) \) to be monotonically increasing, we need:
\[ f'(x) \geq 0 \implies 1 - 4x \geq 0 \]
This simplifies to:
\[ 1 \geq 4x \implies x \leq \frac{1}{4} \]

7. **Identifying the Interval**:
The largest interval where \( f(x) \) is increasing is:
\[ (-\infty, \frac{1}{4}] \]

8. **Conclusion**:
Thus, the answer is:
\[ \text{The largest interval in which } y = f(x) \text{ is monotonically increasing is } (-\infty, \frac{1}{4}]. \]