If `x_1 and x_2` are abscissae of two points on the curve `f(x) = x - x^2` in the interval `(0,1).` then maximum value of the expression `(x_1+x_2)-(x_1^2+x_2^2)` is

Options

A. `(1)/(2)`
B. `(1)/(2)`
C. 1
D. 2

Answer

To solve the problem, we need to find the maximum value of the expression \( (x_1 + x_2) - (x_1^2 + x_2^2) \) where \( x_1 \) and \( x_2 \) are the abscissae of two points on the curve defined by the function \( f(x) = x - x^2 \) in the interval \( (0, 1) \).

### Step-by-step Solution:

1. **Understanding the Expression**:
We can rewrite the expression \( (x_1 + x_2) - (x_1^2 + x_2^2) \) using the identity \( a^2 + b^2 = (a + b)^2 - 2ab \):
\[ (x_1 + x_2) - (x_1^2 + x_2^2) = (x_1 + x_2) - \left((x_1 + x_2)^2 - 2x_1x_2\right) \]
Let \( s = x_1 + x_2 \) and \( p = x_1 x_2 \). Thus, we can rewrite the expression as:
\[ s - (s^2 - 2p) = s - s^2 + 2p \]

2. **Finding the Range of \( s \) and \( p \)**:
Since \( x_1 \) and \( x_2 \) are in the interval \( (0, 1) \):
- The minimum value of \( s \) is \( 0 \) (when both are \( 0 \)) and the maximum value is \( 2 \) (when both are \( 1 \)).
- The product \( p = x_1 x_2 \) will have a maximum when \( x_1 \) and \( x_2 \) are equal. The maximum occurs at \( x_1 = x_2 = \frac{1}{2} \), giving \( p = \frac{1}{4} \).

3. **Finding the Maximum of the Expression**:
We need to maximize \( s - s^2 + 2p \). We can express \( p \) in terms of \( s \):
\[ p = \frac{s^2}{4} \]
Thus, substituting \( p \) into the expression:
\[ E(s) = s - s^2 + 2\left(\frac{s^2}{4}\right) = s - s^2 + \frac{s^2}{2} = s - \frac{s^2}{2} \]

4. **Finding the Critical Points**:
To find the maximum, we take the derivative of \( E(s) \):
\[ E'(s) = 1 - s \]
Setting the derivative to zero:
\[ 1 - s = 0 \implies s = 1 \]

5. **Evaluating the Maximum**:
We check the second derivative to confirm it's a maximum:
\[ E''(s) = -1 < 0 \]
Now, substituting \( s = 1 \) back into the expression:
\[ E(1) = 1 - \frac{1^2}{2} = 1 - \frac{1}{2} = \frac{1}{2} \]

### Conclusion:
The maximum value of the expression \( (x_1 + x_2) - (x_1^2 + x_2^2) \) is \( \frac{1}{2} \).