If `| int_(a)^(b) f(x)dx|= int_(a)^(b)|f(x)|dx,a ltb,"then " f(x)=0` has

Options

A. exactly one root in (a,b)
B. at least one root in (a,b)
C. no root in (a,b)
D. none of these

Explanation

To solve the problem, we need to prove that if

\[ | \int_{a}^{b} f(x) \, dx | = \int_{a}^{b} |f(x)| \, dx \]

and \( a < b \), then \( f(x) = 0 \) has no roots in the interval \( [a, b] \).

### Step-by-Step Solution:

1. **Understanding the given equation**:
The equation states that the absolute value of the definite integral of \( f(x) \) from \( a \) to \( b \) is equal to the integral of the absolute value of \( f(x) \) over the same interval.

2. **Applying properties of integrals**:
From the properties of definite integrals, we know that:

\[ | \int_{a}^{b} f(x) \, dx | \leq \int_{a}^{b} |f(x)| \, dx \]

This inequality holds for any function \( f(x) \).

3. **Equality condition**:
The equality \( | \int_{a}^{b} f(x) \, dx | = \int_{a}^{b} |f(x)| \, dx \) holds if and only if \( f(x) \) does not change sign over the interval \( [a, b] \). This means \( f(x) \) is either non-negative or non-positive throughout the interval.

4. **Analyzing the implications**:
If \( f(x) \) is non-negative, then \( f(x) \geq 0 \) for all \( x \in [a, b] \). If \( f(x) \) is non-positive, then \( f(x) \leq 0 \) for all \( x \in [a, b] \).

5. **Considering roots of \( f(x) \)**:
If \( f(x) = 0 \) has a root in the interval \( [a, b] \), then there exists some \( c \in [a, b] \) such that \( f(c) = 0 \). This would imply that \( f(x) \) changes sign around \( c \) (i.e., it would be positive on one side and negative on the other), contradicting our earlier conclusion that \( f(x) \) does not change sign.

6. **Conclusion**:
Therefore, if \( | \int_{a}^{b} f(x) \, dx | = \int_{a}^{b} |f(x)| \, dx \), it must be the case that \( f(x) = 0 \) has no roots in the interval \( [a, b] \).

### Final Statement:
Thus, we conclude that if \( | \int_{a}^{b} f(x) \, dx | = \int_{a}^{b} |f(x)| \, dx \) for \( a < b \), then \( f(x) = 0 \) has no roots in the interval \( [a, b] \).