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Prove that if $f(x)$ is continuous on $[a,b]$ and $f(a)\lt 0\lt f(b)$ or $f(b)\lt 0\lt f(a)$ , 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:

Answer:

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

Solution:

We have proven both cases here and here.

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

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SAMPLE SOLUTION

GAIN AN ADVANTAGE

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