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Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing funct...
(7) Let R= {f [0,1] - R | f continuous} be the ring of all continuous functions from the interval [0,1] to the real numbers. (a) For cE [0, 1, prove that Me := {feR | f(c) = 0} is a maximal ideal of R. Hint: consider the evaluation map ec- (b) Show that if M is any maximal ideal of R then there exists a cE [0,1 such that M = Me. Hint: show that any maximal ideal M...
23. Let be a function defined and continuous on the closed interval (a,b). If f has a relative maximum at cand a<c<b, which of the following statements must be true? 1. f'(c) exists. II. If f'(c) exists, then f'(c)= 0. III. If f'(c) exists, then f"(c)<0. (A) II only (B) III only (C) I and II only (D) I and III only (E) II and III only
Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a, b]. In class, we proved that f attains its maximum on that interval, i.e. there exists Imar E la, so that f(Imar) > f(x) for all r E (a,b]. We didn't prove that f attains its minimum on the interval, but I claimed that the proof is similar. In fact, you can use the fact that f attains its maximum on any closed interval...
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest interval T where f is strictly increasing. (b) For this interval Z, determine the range f(T) (c) Let T- f(I). Show that the function f : I -» T is injective and surjective. (d) Determine the inverse function f-i : T → 1. (e) Verify that (fo f-1)()-y for every y E...
Let f be defined on an open interval I containing a point a (1) Prove that if f is differentiable on I and f"(a) exists, then lim h-+0 (a 2 h2 (2) Prove that if f is continuous at a and there exist constants α and β such that the limit L := lim h2 exists, then f(a)-α and f'(a)-β. Does f"(a) exist and equal to 2L? Let f be defined on an open interval I containing a point a...
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
Suppose that f' exists and is continuous on a nonempty open interval (a,b) with f'(c) + 0 for all 2 € (a,b). | Prove that f is one-to-one on (a, b) and that f((a,b)) is an open interval II: if (c,d) is the open interval from (i), show that f-1EC'((c,d)), i.e. f-1 has a continuous first derivative on (a, b).
A function f:R HR is said to be strictly increasing if f(x1) < f(12) whenever I] < 12. Prove: If a differentiable function f is strictly increasing, then f'(x) > 0. Then give counterexamples to show that the following statements are false, in general. (i) If a differentiable function f is strictly increasing, then f'(2) >0 for all 1. (ii) If f'(x) > 0 for all x, then f is strictly increasing -
Problem I. Let f: R -> R be any map and suppose that the graph Tf CR is closed and connected. Show that f is continuous. Problem I. Let f: R -> R be any map and suppose that the graph Tf CR is closed and connected. Show that f is continuous.
7. [8 POINTS] Let f: R → R be a strictly increasing function. Prove by way of contradiction that there cannot be more than one place where f crosses the x-axis.