1. ## integral/continuety proof question..

function f(x) continues on [a,b]
suppose that for every sub part $[\alpha ,\beta ]\subseteq [a,b]$
we have $\int_{\alpha}^{\beta}f(x)dx>0$.
prove that f(x)>=0 for $x\in[a,b]$

if its wrong give a contradicting example??

i dont have a clue from here to start or how to go.
from the given i can conclude that if the sum of all subsections gives us a positive result then
the total sum from a to b has to positive too.inte

2. Originally Posted by transgalactic
function f(x) continues on [a,b]
suppose that for every sub part $[\alpha ,\beta ]\subseteq [a,b]$
we have $\int_{\alpha}^{\beta}f(x)dx>0$.
prove that f(x)>=0 for $x\in[a,b]$
Can you prove this theorem: If a function, $f$, is continuous on $[a,b]$ and $\left( {\exists c \in [a,b]} \right)\left[ {f(c) < 0} \right]$
then there is a subinterval on which $f$ is non-positive?

3. i dont know how to prove it
any hint
??

4. i tried to prove it by contradiction:
suppose
there is a section which has negative value
$
\int_{\alpha}^{\beta}f(x)dx<0
$

then
$
\int_{a}^{b}f(x)dx<0
$

that as far as i can think of