Chapter 6.1, Problem 29ES

Determine whether the statement is true or false. Explain your answer. [In each exercise, assume that $f$ and $g$ are distinct continuous functions on $\left[a,b\right]$ and that $A$ denotes the area of the region bounded by the graphs of $y=f\left(x\right),y=g\left(x\right),x=a$ , and $x=b$ .]

If

${\displaystyle {\int }_a^b\left[f\left(x\right)-g\left(x\right)\right]dx}=0$

then the graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$ cross at least once on $\left[a,b\right]$ .