What is the difference between Fermat's Theorem and the Extreme Value Theorem?

Question

In the middle of proving Extreme Value Theorem so those studying at an undergraduate level can understand it, and can't help notice the similarity between theorems. Can someone please explain the difference between them.

Answer 1

The extreme value theorem is just about continuity - it says that continuous functions achieve maximal and minimal values on (finite) closed intervals (in their domains).

Fermat's theorem says something about what happens at those extreme values, from the point of view of differentiability: it says that the derivative at a point where an extreme value is achieved is always zero.

• ... or undefined.

• ... or we're looking at an endpoint of the interval, not inside the interval.

Specifically, we have:

Suppose $f$ is a continuous function defined on $[a,b]$. We know that $f$ has an extreme value here by EVT. Suppose $f$ attains an extreme value at $c\in [a,b]$. Then either $c=a$, or $c=b$, or $f'(c)=0$, or $f'(c)$ is undefined.

So Fermat's theorem tells us a bit more about the (points where the) extreme values of $f$ on $[a,b]$ (occur), while the EVT tells us just that they exist in the first place.