If the area bounded by `x`-axis, the curve `y = f(x)` and the ordinates `x = c and x = d` is independent of `d,AA d > c` (`c` is constant), then f is

Options

A. (a)A non-zero constant function
B. (b)Identify function
C. (c)Zero function
D. (d)Parabolic function

Answer

To solve the problem, we need to find the function \( f(x) \) such that the area bounded by the x-axis, the curve \( y = f(x) \), and the ordinates \( x = c \) and \( x = d \) is independent of \( d \) (where \( d > c \) and \( c \) is a constant).

### Step-by-Step Solution:

1. **Understanding the Area**:
The area \( A \) bounded by the x-axis, the curve \( y = f(x) \), and the lines \( x = c \) and \( x = d \) can be expressed as:
\[ A = \int_{c}^{d} f(x) \, dx \]

2. **Independence from \( d \)**:
Since the area is independent of \( d \), it implies that changing \( d \) does not change the value of the area \( A \). This means that the integral must yield a constant value regardless of the upper limit \( d \).

3. **Differentiating the Area**:
To analyze this, we can differentiate the area with respect to \( d \):
\[ \frac{dA}{dd} = \frac{d}{dd} \left( \int_{c}^{d} f(x) \, dx \right) = f(d) \]
For the area to be independent of \( d \), the derivative \( \frac{dA}{dd} \) must equal zero:
\[ f(d) = 0 \]

4. **Conclusion about \( f(x) \)**:
Since \( f(d) = 0 \) for any \( d > c \), it follows that \( f(x) \) must be zero for all \( x \) in the interval \( [c, \infty) \). Therefore, the function \( f(x) \) must be the zero function:
\[ f(x) = 0 \quad \text{for all } x \]

5. **Final Answer**:
The function \( f(x) \) is:
\[ f(x) = 0 \]