other theorems

Bolzano-Weierstrass’s Theorem: If f is continuous in [a,b], then f has at least one absolute maximum and one absolute minimum in the interval. That is

 

Intermediate-value Theorem: If f is continuous in [a,b] and

 

  Demo. let g(x) = f(x) – k →g is continuous in a, and

  then 

                                                                (Bolzano's Theorem)

  In every case

 

 

 

 

Exercise: Demonstrate that exists a value c in (2,5) such that f(c) = 20/3, if

 

 

 

 

Solution: Use the Intermediate-value Theorem in this interval.