The Extreme Value Theorem

Extreme Value Theorem (EVT)

Continuous functions on a closed interval are guaranteed to have a maximum and a minimum.

Extreme Value Theorem

If $f\left(x\right)$ is a continuous function on $\left[a,b\right]$, then $f\left(x\right)$ must have both a maximum and minimum value on $\left[a,b\right]$.

Formally, there exists$m,M\in[a,b]$ such that $f(M)\ge f(x)\ge f(m) \text{, for all}\ x\in[a,b]$