Determine Whether Each of the Following Statements Is True or False

Essay

Determine whether each of the following statements is true or false. If the statement is true, provide an appropriate justification. If the statement is false, provide a counterexample.(a) If $f ( x ) \neq 0 \text { on } [ a , b ] \text { then } \int _ { a } ^ { b } f ( x ) d x \neq 0$ (b) If $f$ is continuous on an interval containing a,b, and c, then $\int _ { a } ^ { b } f ( x ) d x = \int _ { a } ^ { c } f ( x ) d x - \int _ { b } ^ { c } f ( x ) d x$ (c) If $\int _ { a } ^ { b } f ( x ) d x \geq 0 , \text { then } f ( x ) \geq 0 \text { on } [ a , b ]$