- #1

- 16

- 0

## Homework Statement

f,g are cont. fcns on [a,b] and [tex]\int f[/tex] [tex]\geq[/tex] [tex]\int g[/tex] for any subinterval [i,j] of [a,b]. Show f(x) [tex]\geq[/tex] g(x) on [a,b].

## Homework Equations

Don't know if I should have used something like the fundamental theorem or mean value of integrals or something. I was using theorems like those for other problems in this section but wasn't sure about this one so I took a different direction.

## The Attempt at a Solution

Assume that this isn't true, i.e., there exists some c in [a,b] s.t. g(c) > f(c).

f and g are continuous so let's choose our [tex]\epsilon[/tex] = g(c) - f(c), so there should exist some [tex]\delta[/tex] > 0 where:

|f(x) - f(c)| < [tex]\epsilon[/tex] whenever |x - c| < [tex]\delta[/tex]

Thus 2f(c) - g(c) < f(x) < g(c) when x is in (c - [tex]\delta[/tex], c + [tex]\delta[/tex])

So f(x) < g(c) whenever x is in (c - [tex]\delta[/tex], c + [tex]\delta[/tex]).

So consider any partition p of any subinterval [i,j] of [a,b] where the one of the subintervals of [i,j] is (c - [tex]\delta[/tex], c + [tex]\delta[/tex]).

I stopped here because I'm pretty sure I chose the wrong epsilon and I wasn't really sure if I was even in the right direction.