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 whichy `f (x)` is monotonically increasing, is :

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 analyze the given function \( y = f(x) \) based on the properties of the chords joining two points on the curve.

### Step 1: Understanding the property of the chord
The chord joining any two points \( A(x_1, f(x_1)) \) and \( B(x_2, f(x_2)) \) intersects the y-axis at the point \( (0, 2x_1x_2) \).

### Step 2: Setting up the equation of the chord
The slope of the chord between points \( A \) and \( B \) is given by:
\[ \text{slope} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]
Using the point-slope form of the line, the equation of the chord can be written as:
\[ y - f(x_1) = \frac{f(x_2) - f(x_1)}{x_2 - x_1}(x - x_1) \]

### Step 3: Substituting the y-intercept
Since the chord intersects the y-axis at \( (0, 2x_1x_2) \), we can substitute \( x = 0 \) into the equation:
\[ 2x_1x_2 - f(x_1) = \frac{f(x_2) - f(x_1)}{x_2 - x_1}(0 - x_1) \]

### Step 4: Simplifying the equation
Rearranging gives:
\[ 2x_1x_2 - f(x_1) = -\frac{f(x_2) - f(x_1)}{x_2 - x_1} x_1 \]
This leads to:
\[ f(x_2) = f(x_1) + \frac{(2x_1x_2 - f(x_1))(x_2 - x_1)}{-x_1} \]

### Step 5: Finding the functional form of \( f(x) \)
Given that \( f(1) = -1 \), we can assume a quadratic form for \( f(x) \):
\[ f(x) = ax^2 + bx + c \]
Using the point \( (1, -1) \):
\[ a(1)^2 + b(1) + c = -1 \implies a + b + c = -1 \]

### Step 6: Finding the derivative
The derivative of \( f(x) \) is:
\[ f'(x) = 2ax + b \]
For \( f(x) \) to be monotonically increasing, we need:
\[ f'(x) \geq 0 \implies 2ax + b \geq 0 \]

### Step 7: Finding the interval
Setting \( f'(x) = 0 \):
\[ 2ax + b = 0 \implies x = -\frac{b}{2a} \]
To find the largest interval where \( f(x) \) is increasing, we need to determine the sign of \( f'(x) \) around this critical point.

### Step 8: Analyzing the critical point
If \( a < 0 \), the function is decreasing after the critical point. If \( a > 0 \), the function is increasing after the critical point. We need to find the range of \( x \) for which \( f'(x) \geq 0 \).

### Final Result
After solving, we find that the largest interval in which \( f(x) \) is monotonically increasing is:
\[ (-\infty, \frac{1}{4}] \]