Extreme Value Theorem

Theorem(Weierstrass) If $f(x)$ is a continuous function on a closed interval $[a,b]$, then $f(x)$ attains its local extreme values on $[a,b]$.

Intermediate Value Theorem

Theorem If a function $f(x)$ is continuous on a closed interval $[a, b]$ and $L$ is any number between $f(a)$ and $f(b)$, where $f(a) \neq f(b)$, then there exists at least one number $c$ in the open interval $(a, b)$ such that $f(c) = L$.

The most common and useful scenario for applying the Intermediate Value Theorem (IVT) is to find a zero of a continuous function. If $f(a)f(b)<0$ and $f$ is continuous on $[a,b]$, then there exists a number $c$ such that $f(c)=0$.