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) – kg is continuous in a, and


                                                                (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.

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