MATH 205 Lecture Notes - Lecture 16: Mean Value Theorem, Thx, Maxima And Minima
Document Summary
Rolle"s theorem let f be a function that satisfies the. 1 f is continuous on the closed interval the open interval a b aib f is differentiable on. 3 fca then there is a number c f b in ca b such that f"cc. Proof for rolle"s theorem 3 cases fly k a constant then thx. I i can be taken at any number fcx. Fca fcb c maximum at at it mustattain this maximum value the open interval aib then f has a f. 0 by fermat"s theorem c and by hypothesis 2 c therefore f"cc in so the number. Since can apply by hypotheses at a number local is differentiable ex: 1 a. 3 aib f has a minimum value in it attains this minimum value at. Fermat"s theorem a b by extreme value theorem fcb a number and since fca in ca b c ex: 1 a. 1 b x"tx 1 0 has exactly real.
