Please solve exercise 3.3.3 with detailed explanations • Exercise 3.2.10 in Section 3.2: Let f: R → R and g: R → R be continuous functions. Suppose that f(r) = g(r)for every rational number r. Prove that f(x) = g(x) for every real number î.• Exercise 3.3.3 in Section 3.3: Suppose ƒ: (0, 1) → R is a continuous function such that the limits lim f(x) andx →0lim f(x) both exist and equal 0. Prove that f attains an absolute minimum or an absolute maximum on thex→1interval (0, 1).As usual, "or" is inclusive: there might be a minimum but no maximum, a maximum but no minimum, or both aminimum and a maximum.RemarksThe point of Exercise 3.2.10 is that a continuous function on R is completely determined by the values on a suitablecountable subset of the domain, namely, a countable dense subset. (More generally, a topological space is calledseparable when there exists a countable dense subset.)In Exercise 3.3.3, you could also write the indicated limits as the one-sided limits lim f(x) and lim f(x). By thex→1x →0+author's definition of limit, however, the variable x is automatically restricted to the declared domain of the function,so the decorations + and are optional here.

Ans-We are given a condition that f: (0, 1)→ R is a continuous function such that the limits lim…

• Exercise 3.3.3 in Section 3.3: Suppose f: (0, 1) → R is a continuous function such that the limits lim f(x) and x→0 lim f(x) both exist and equal 0. Prove that f attains an absolute minimum or an absolute maximum on the x→1 interval (0, 1). As usual, "or" is inclusive: there might be a minimum but no maximum, a maximum but no minimum, or both a minimum and a maximum.

• Exercise 3.3.3 in Section 3.3: Suppose f: (0, 1) → R is a continuous function such that the limits lim f(x) and x→0 lim f(x) both exist and equal 0. Prove that f attains an absolute minimum or an absolute maximum on the x→1 interval (0, 1). As usual, "or" is inclusive: there might be a minimum but no maximum, a maximum but no minimum, or both a minimum and a maximum.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.3: The Natural Exponential Function

Problem 52E

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