Prove that if $f(x)$ is continuous on $[a,b]$ and $f(b)\lt 0\lt f(a)$, then there exists a number $K$ between $a$ and $b$ such that $f(K)=0$.

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Lesson Parent: 
Function Limits VI: Continuous Functions
Problem: 

Prove that if $f(x)$ is continuous on $[a,b]$ and $f(b)\lt 0\lt f(a)$, then there exists a number $K$ between $a$ and $b$ such that $f(K)=0$.

Answer: 

SAMPLE SOLUTION

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Solution: 

Because $f(x)$ is continuous at $a$ and <Sign in to see all the formulas> we know that there is an interval <Sign in to see all the formulas> where $f(x)$ is positive. (We know this from this problem where we can choose <Sign in to see all the formulas> and $b$ to be small enough that <Sign in to see all the formulas> is above zero.) Similarly, I also know that there is some interval <Sign in to see all the formulas> where <Sign in to see all the formulas>K<Sign in to see all the formulas>\delta<Sign in to see all the formulas>f(x)<Sign in to see all the formulas>[a,\delta)<Sign in to see all the formulas>K\lt b<Sign in to see all the formulas>f(K)=0<Sign in to see all the formulas>\mbox{lub} \{\delta: f(x)\gt 0\mbox{ on }[a,\delta)\}<Sign in to see all the formulas>f(x)<Sign in to see all the formulas>x<Sign in to see all the formulas>K<Sign in to see all the formulas>f(b)\lt 0<Sign in to see all the formulas>(\phi,b]<Sign in to see all the formulas>f(x)<Sign in to see all the formulas>K\neq b<Sign in to see all the formulas>a\lt K\lt b<Sign in to see all the formulas>f(K)<Sign in to see all the formulas>x<Sign in to see all the formulas>K<Sign in to see all the formulas>f(x)<Sign in to see all the formulas>K<Sign in to see all the formulas>f(K)<Sign in to see all the formulas>K<Sign in to see all the formulas>f(K)<Sign in to see all the formulas>K<Sign in to see all the formulas>f(x)<Sign in to see all the formulas>[a,b]<Sign in to see all the formulas>f(K)<Sign in to see all the formulas>a\lt K\lt b<Sign in to see all the formulas>f(K)=0$.