integral/continuety proof question..

• Feb 19th 2009, 12:45 PM
transgalactic
integral/continuety proof question..
function f(x) continues on [a,b]
suppose that for every sub part $\displaystyle [\alpha ,\beta ]\subseteq [a,b]$
we have$\displaystyle \int_{\alpha}^{\beta}f(x)dx>0$.
prove that f(x)>=0 for $\displaystyle x\in[a,b]$

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
• Feb 19th 2009, 03:16 PM
Plato
Quote:

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

Can you prove this theorem: If a function, $\displaystyle f$, is continuous on $\displaystyle [a,b]$ and $\displaystyle \left( {\exists c \in [a,b]} \right)\left[ {f(c) < 0} \right]$
then there is a subinterval on which $\displaystyle f$ is non-positive?
• Feb 19th 2009, 11:37 PM
transgalactic
i dont know how to prove it
any hint
??
• Feb 24th 2009, 10:16 AM
transgalactic
i tried to prove it by contradiction:
suppose
there is a section which has negative value
$\displaystyle \int_{\alpha}^{\beta}f(x)dx<0$
then
$\displaystyle \int_{a}^{b}f(x)dx<0$
that as far as i can think of