16. For y = f(x), if
d²y⁄dx² = 0,
dy⁄dx = 0 at x = 0,
and y = 1 at x = 1, the value of y at x = 2 is ________.
Problem Overview
We’re given a function y = f(x) satisfying the following conditions:
- ( d^2y/dx^2 = 0 )
- ( dy/dx = 0 ) at ( x = 0 )
- ( y = 1 ) at ( x = 1 )
We need to compute the value of y at x = 2.
Step-by-Step Solution
1. Evaluate the Second Derivative
The condition ( d^2y/dx^2 = 0 ) implies:
( dy/dx = C_2 )
So the first derivative is constant.
2. Apply the Slope Condition
Given (dy/dx = 0 ) at ( x = 0 ), we have:
( C_2 = 0 )
Thus the slope is zero everywhere.
3. Integrate to Find y(x)
Integrating gives:
( y = C_1 )
The function is constant.
4. Apply y(1) = 1
Using the point ( y = 1 ) at ( x = 1 ):
( C_1 = 1 )
This gives:
( y(x) = 1 ) for all x
Final Answer
y(2) = 1
Common Options Explained
| Option | Verdict | Explanation |
|---|---|---|
| y = 0 | Incorrect | Contradicts the condition y = 1 at x = 1 |
| y = 1 | Correct | The function is constant due to zero slope and second derivative |
| y = 2 | Incorrect | Requires non-zero slope, violating dy/dx = 0 |
| Cannot be determined | Incorrect | Conditions uniquely determine y(x) = 1 |
