# Sup problem

1. Oct 15, 2009

### andilus

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

3. The attempt at a solution

Last edited: Oct 15, 2009
2. Oct 15, 2009

### rasmhop

Start by showing that there is one x in [a,b] such that f(x) = 0. Then form the set
$$S = \{x \in [a,b] | f(x) = 0\}$$
You have already shown that S is non-empty and you know that it's bounded, so it must have a supremum. Let
$$x_0 = \sup\,S$$
Since $x_0$ is an upper bound for S, if we can show $x_0 \in S$ we have shown that it's the largest element in S. So all you need to do is show $x_0 \in S$. The easiest way to do this is to assume $x_0 \notin S$, i.e. $y_0 = f(x_0) \not= 0$. Now since f is continuous at $x_0$ we can find some $\delta > 0$ such that if $x \in (x_0-\delta,x_0 + \delta)$ then $f(x) \in (0,2y_0)$ (let $\epsilon = |y_0|$). Now $x_0 - \delta$ is an upper-bound for S, but less than $x_0$.

3. Oct 19, 2009

### andilus

but how to prove x0 is in [a,b]?

4. Oct 19, 2009

### andilus

i seem to know...
if x0<a, obviously wrong.
if x0>b, we can find some x such that b-$$\delta$$<x<b satisfying that x is upper bound of the set.