Prove the Critical Point Theorem

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Lesson Parent: 
Geometry of Functions VI: Critical Points
Problem: 

Prove that if $f(x)$ has a maximum or minimum at $e$, then $f\,'(e)=0$ or $f\,'(e)$ does not exist.

Answer: 

It is true that if $f(x)$ has a maximum or minimum at $e$, then $f\,'(e)=0$ or $f\,'(e)$ does not exist.

SAMPLE SOLUTION

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Solution: 

The proof is by contraposition. In this type of proof we assume that <Sign in to see all the formulas> exists and <Sign in to see all the formulas> and then prove that $f(x)$ does not have a maximum or minimum at $e$. We have two cases to consider. The first is when <Sign in to see all the formulas> and the second is when <Sign in to see all the formulas>.

When <Sign in to see all the formulas>, we know that for all $k$ sufficiently small that <Sign in to see all the formulas>. By looking at the definition of maximum and minimum we see that $f(e)$ cannot be a maximum or minimum.

Similarly, when <Sign in to see all the formulas>, we know that for all $k$ sufficiently small that <Sign in to see all the formulas>. By looking at the definition of maximum and minimum we see that $f(e)$ cannot be a maximum or minimum.

The proof is now complete.