- #1

- 775

- 1

## Main Question or Discussion Point

Consider the function [itex]F(x)[/itex] where [itex]F(x) > 0 [/itex] for all [itex]x[/itex].

If we know that [tex]\int^{x_{2}}_{x_{1}}F(x)dx = 0 [/tex] can we prove that [itex]x_{1}=x_{2}[/itex] ?

I can visually imagine that they are equal since the function is always positive, its integral must be monotically increasing, but I can't imagine how I would prove this.

I made the problem myself while studying probability so I'm not sure a solution exists. If a solution does not exist I'd like to see a counterexample.

I would imagine that the solution employs the MVDT and FTC, but as I mentioned before, I'm not good at actually writing the statements for proofs so I need some help here.

BiP

If we know that [tex]\int^{x_{2}}_{x_{1}}F(x)dx = 0 [/tex] can we prove that [itex]x_{1}=x_{2}[/itex] ?

I can visually imagine that they are equal since the function is always positive, its integral must be monotically increasing, but I can't imagine how I would prove this.

I made the problem myself while studying probability so I'm not sure a solution exists. If a solution does not exist I'd like to see a counterexample.

I would imagine that the solution employs the MVDT and FTC, but as I mentioned before, I'm not good at actually writing the statements for proofs so I need some help here.

BiP