Chapter 6.3, Problem 46E

To determine

The given statement is true or false.

Answer to Problem 46E

The given statement is False.

Explanation of Solution

Given information:

  ${{\displaystyle \int }}_a^{b}f(x)dx=F(b)-F(a)$

Since,

  ${{\displaystyle \int }}_a^{b}f(x)dx=F(b)-F(a)$

Where F is any anti-derivative of f

Now,

  ${{\displaystyle \int }}_a^{b}f(x)dx=0$

So,

  $F(b)=F(a)$

It does not mean that-

  $f(b)=f(a)$

It may be true.

So, if

  ${{\displaystyle \int }}_a^{b}f(x)dx=0$

Then, it may or may not be

  $f(a)=f(b)$