Extreme value theorem
Definition
Suppose is a continuous function on a closed interval (note that may be defined on a bigger domain, but we are interested in the restriction of to the closed interval and require it to be continuous). Then, attains its minimum and maximum value on the interval. In other words, the following statements are true:
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
- Existence of maximum value: There is a real number such that for all and there exists such that .
- Existence of minimum value: There is a real number such that for all and there exists such that .
![{\displaystyle x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/026357b404ee584c475579fb2302a4e9881b8cce)
![{\displaystyle c_{1}\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/351a9c6cac2f733a6e6cdabc2f0d8363644ee6d6)
![{\displaystyle c_{2}\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/255bf496bbf61c38e0f32d187442a94f971a81e1)
