i need part c and d onlyurgently please do this (2) (a) Consider a function f, defined in some open neighbourhood of xo € R,i.e., f : (a; b) → R, a < ro < b. Suppose f is continuous on (a; b)and f(ro) + 0. Show that there is some open inteval around ro, e.g.,ro € (a1; b1) C (a; b), such that f is non-vanishing on (a1;b1); that is,f(x) +0 for all a1

Answered: Image /qna-images/answer/463d46d0-9fc3-4938-9eba-3a15784f317f.jpg

(c) Give two (essentially different) examples of functions f : (a; b) → R, which are differentiable, but whose derivative is not continuous. (d) Suppose f: (a; b) → R is differentiable and its derivative f' : (a; b) → R is continuous. Suppose also that f'(xo) # 0 for some ro € (a; b). Conclude that there exists an interval (a1; b1) around ro and an in- terval (c; d) around f(ro), such that f : (a1;b1) → (c; d) is one-to-one and onto (i.e., bijection).

(c) Give two (essentially different) examples of functions f : (a; b) → R, which are differentiable, but whose derivative is not continuous. (d) Suppose f: (a; b) → R is differentiable and its derivative f' : (a; b) → R is continuous. Suppose also that f'(xo) # 0 for some ro € (a; b). Conclude that there exists an interval (a1; b1) around ro and an in- terval (c; d) around f(ro), such that f : (a1;b1) → (c; d) is one-to-one and onto (i.e., bijection).

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.3: Rates Of Change And Behavior Of Graphs

Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...

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