# Sup problem

if f is continuous on [a,b] with f(a)<0<f(b), show that there is a largest x in [a,b] with f(x)=0

i think it can be done by least upper bounds, but i dun know wat is the exact prove.

Look up "Intermediate Value Theorem" or "Bolzano's Theorem."

LCKurtz
Homework Helper
Gold Member
As another poster suggested, the intermediate value theorem guarantees there is an x in [a,b] where f(x) = 0. And your idea of using the lub is a good one. So let

$z =$ lub $\{ x \in [a,b] | f(x) = 0\}$

So what you need to show to finish the problem is:

1. z is in [a,b]
2. f(z) = 0
3. No value x > z in [a,b] satisfies f(x) = 0.

Last edited:
LCKurtz