Prove the Extreme-Value Theorem.

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Lesson Parent: 
Geometry of Functions II: The Extreme-Value Theorem
Problem: 

Prove the Extreme-Value Theorem.

Answer: 

SAMPLE SOLUTION

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Solution: 

The Extreme-Value Theorem states that if $f(x)$ is continuous on $[a,b]$, then $f(x)$ takes on both maximum and minimum values on $[a,b]$. We have proven the maximum case here and the minimum case here. The two proofs are nearly identical so you only need to understand one of them to understand both.