Question

True or False If \(\int _ { a } ^ { b } f ( x ) d x = 0 ,\) then \(f ( a ) = f ( b ) .\) Justify your answer.

Short answer

False

Step-by-step solution

Statement Analysis

Understand the statement. The problem is asking whether the definite integral of a function being zero over an interval implies that the function value at the endpoints of the interval is the same.

Counter Example

To disprove, construct a counter function where the integral of the function between two points \(a\) and \(b\) is zero, but the function values at a and b are not the same. Consider the function \(f ( x ) = x\) on the interval \([- 1, 1]\). Then the integral is \(\int _{- 1}^{1} x d x = \left[ \frac { 1 }{2} x ^ { 2 } \right]_{- 1}^{1} = 0\). However, \(f( - 1 ) = - 1\) and \(f ( 1 ) = 1\), which are different.

Conclusion

As the function on the interval [-1, 1] disproves the statement, the statement 'If \(\int _ { a } ^ { b } f ( x ) d x = 0 ,\) then \(f ( a ) = f ( b ) \)' is False.