(3) (LATEX question) Give a proof of the second part of the Extreme Value Theorem, 6.3.5, that is, if f : [a, b] → R is continuous then f is bounded below and attains a minimum.

Q: Let f (x, y) = 1 5+√x²+y²-1 Then range of f is:

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Q: What is the global minimum of the function, x² 5 f(x) 3 + 2 on the interval [-2, 2] ? a) 2 b) 23 3…

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Q: 3. Determine whether each of these functions from is one-to-one/onto. (a) f : R → R, f(x) = 3r. (b)…

A: A function f is one-to-one if every element of the range of g corresponds to exactly one element of…

Q: Let ƒ : R\{1} →→ R\{1} be defined by f(x) = x/(x - 1). Show that ƒ is bijective and find f-¹. Also,…

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Q: 4. Let X an Y be sets with A, B C X and let f: X→Y be a function. Recall that {f(x) EYE A}, and…

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Q: Determine whether cach of these functions from Z to Z is one-to-one. a) f(n)=n- I c) f(n)=n b) f(n)…

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve first three…

Q: Let φ1(x) = 1, φ2(x) = x/|x|, φ3(x) = x^2 − 1/3 for −1 ≤x ≤1. Show that (φ1,φ2,φ3) form an…

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Q: Let the function ƒ : [0, 1] →R be defined by { if x = ! (n = 1, 2, 3, · · · ) f(x) = n 1– x2…

A: Given :  f : [0, 1] →ℝ  defined by ;              fx = ex2,if  x=1n    (n=1, 2, 3, ...…

Q: Let f : A → B be a function. Prove that if C C A and D C A and f is injective, then f(C) = 7 (D) =→…

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Q: Exercise 2. Let f : A → B be a function with Y1 and Y2 subsets of B. Prove that if Y1 C Y2, then…

A: Let f:A→B be a function and let Y1,Y2⊆B. Suppose that Y1⊆Y2 To show f-1(Y1)⊆f-1(Y2) Let x∈f-1(Y1)…

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F closed?

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Q: I. Samples – Study these sample functions and their descriptions and fill in the blanks. a) 31 b) 31…

A: Answer of sample (A):