Prove the Intermediate-Value Theorem

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Lesson Parent: 
The Intermediate-Value Theorem
Problem: 

Prove the Intermediate-Value Theorem. We assume that f(x) is continuous on [a,b] and that f(a)≠f(b). Prove that for every number Ω between f(a) and f(b) that there is at least one number ω in (a,b) such that f(ω)=Ω.

SAMPLE SOLUTION

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Solution: 

We define <Sign in to see all the formulas> as an arbitrary value between f(a) and f(b). When we say that <Sign in to see all the formulas> is between f(a) and f(b) we can have either <Sign in to see all the formulas> or <Sign in to see all the formulas>. This means we have to prove the intermediate-value theorem for both of these scenarios.

Let's define <Sign in to see all the formulas>. Since f(x) is continuous on [a,b], we know that e(x) is continuous on [a,b].

Now when <Sign in to see all the formulas> we can subtract <Sign in to see all the formulas> to see that <Sign in to see all the formulas>. When <Sign in to see all the formulas> we can subtract <Sign in to see all the formulas> to see that <Sign in to see all the formulas>.

Since e(x) is continuous on [a,b] and that <Sign in to see all the formulas> or <Sign in to see all the formulas> then there is a number <Sign in to see all the formulas> between a and b such that <Sign in to see all the formulas>. Since <Sign in to see all the formulas> then <Sign in to see all the formulas>.