Extreme Value Theorem (EVT)

Statement of the theorem

Thm: If $f$ is continuous on a closed and bounded interval $[a,b]$, then $\exists \ y \in [a,b]$ such that $\forall \ x \in [a,b], f(y) \geq f(x)$.

Assumptions

As before, if the interval fails to be closed, then, for example, $f(x)=x^2$ fails to attain a maximum on $(0,1)$, for if you give me any $y \in (0,1)$, then $f(\frac{y+1}{2}) > f(y)$.

To see the proofs, read on!

(Video) Extreme Value Theorem

by learnifyable

Here's a video by learnifyable explaining the Extreme Value Theorem.