Extreme value theorem

Definition

Suppose $f$ is a continuous function on a closed interval $[a,b]$ (note that $f$ may be defined on a bigger domain, but we are interested in the restriction of $f$ to the closed interval and require it to be continuous). Then, $f$ attains its minimum and maximum value on the interval. In other words, the following statements are true:

1. Existence of maximum value: There is a real number $M$ such that $f(x)\le M$ for all $x\in [a,b]$ and there exists $c_1\in [a,b]$ such that $f(c_1)=M$.
2. Existence of minimum value: There is a real number $m$ such that $f(x)\ge m$ for all $x\in [a,b]$ and there exists $c_2\in [a,b]$ such that $f(c_2)=m$.

Related facts

Applications

• Rolle's theorem
• Lagrange mean value theorem