7. Which of the following function satisfy Rolles theorem in [-2, 2] interval a) I b) X-1 c) x d) None of these

Q: b. Prove that coth² x-1=csch?x

A: Sinh(x)= (e^x-e^-x)/2. Cosh(x)=(e^x+e^-x)/2

Q: A relation R defined on the set of real numbers is such that xRy and yRx = x = y: R is said to be A…

A: There are many types of relations Symmetric Asymmetric Reflexive Transitive

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Q: A relation Ris defined on N by a R b if a/b e N. For c, d e N, under what conditions is c R1 d?

A: An inverse relation is the set of ordered pairs obtained by interchanging the first and second…

Q: Show that |qbu)|&x + 7元? the closed on interval [-元,T」 fo1 -π2 2くだ gluen that glven gux) for

A: Given g (x) = x +x² To find the maximum value of g(x). We can differentiate the function and equate…

Q: ) Prove that A° U B° C (AU B)°. Give an example to show that equality does not always hold.

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Q: 1. Use the Intermediate Value Theorem to prove that the equation 3x - 4x? = 3 is solvable on the…

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Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x € R: 1/x e A}. 1…

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

A: Let (M,d) be a metric space and let S⊆M. A point a∈S is said to be an Interior Point of S if there…

Q: 2. Let f be defined on an open interval (a, b) and assume that for some e e (a, b) we have f'(c) > 0…

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Q: If X and Y are finite sets, prove that XY = |X|Y. Here, XY is the set of all functions from Y to X.

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Q: يثبت The Cauchy-Schwarz inequality states that (f, g)| ≤ ||f||||g|| for every f, g ¤ H.

A: Using the property of inner product

Q: Let X ~ N(0, 1). For any € > 0, show that P(|X| > c) ≤ ²c - ??

A: Note: Hi there! Thank you for posting the question. As you have posted multiple questions, as per…

Q: Let A, B, C, and D be sets.

A: First, (AxC)U(BXD) ⸦(AUC) x(BUD)Take left hand side first , let  (x,y) ∈(AxC) U(BxD)=> (x,y)…

Q: Show that the function f(z) = from [0, 0) to [0, ∞) has a fixed point c. Hint: Set f(r) = z and show…

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Q: ) suppose fle) oa defferentialle funchou on the intervel E-7,1J such that H-7) =4 and fl)=1. Explain…

A: SOLN 3B OKLY AS ASKED

Q: T is a function?

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Q: 4. Let A C R" be both connected and sequentially compact. Prove that if F: A –→ R is continuous,…

A: Consider the given information.

Q: Show that the intervals S [1,3] and T =[2,8] are equinumerous by finding a. bijection between them.

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Q: Prove than any pair of elements in H can be mapped to the positive y-axis by an element of SL(2, R).

A: According to the given information, it is required to prove that any pair of elements in H can be…

Q: Sketch the set of points satisfying: |z - 1 + 2i| = |z + 3 + i|

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Q: ☺ show dhat 2) G(x,x) => Uola) Uorlo) were G is green funcfion

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Q: Define the relation ~ on the set R of real numbers by x~y if and only if x = -y. Show that ~ is not…

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Q: Show that f: R → R defined as f(a) = 3a³- 4 is a one to one %3D iuicuon.

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Q: Prove that interval [0,1] is uncountable.

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Q: b( Let S be the relation defined on P ((1, 2, 3, 4}) defined by X SY if and only if IX| = |Y| (mod…

A: The complete solutions are given below

Q: if f is measurable then - f is messlurable

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Q: 9:Find the absolute exlemum Verlues of flx) x²-4x+24 over the interNal Cl,41

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Q: Prove or disprove the relation C on Z defined by C = {(x, y): x=y (mod 5)} satisfies the two…

A: I just used the definition of congruence and divisibility

Q: 6. Is the function h: Z → Z defined by ( 2n if n 20 h(n) -n if n < 0 one-to-one? Is it onto?

A: The solution is given as follows

Q: 4. Show that the intervals S =[1,3] and T =[2,8] are equinumerous by finding a bijection between…

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Q: For p = (1, 0) and x = (x₁,x2,...,xn) ER", define ||2|p:= - ². |P Show that n jyjx|py|q for x, y…

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Q: Prove that if A is not connected, then A is not an interval.

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Q: Theorem 5 there is auniqe Reul for X=2 Sit X ER

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Q: Show that the class of closed interval [a,b), where a is rational and b is irrationals and a <b, is…

A: Base for topology: A collection B of subsets of topological space X is said to be base for topology…

Q: Show that if ƒ′′&gt; 0 throughout an interval [a, b], then ƒ′ has at most one zero in [a, b] . What…

A: Given:

Q: 20. Give two examples of functions from Z to Z that are one- to-one but not onto.

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Q: The relation xy >= 1 where x,y are on the set of all integers is reflexive. True False

A: Given relation xy≥1, where x,y are on the set of all integers

Q: |A:k||2||Br:||2) < ||A||F||B||F k=1

A: answer is in next step

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A: True

Q: 4.Describe the largest set S on which it is correct to say that f is continuous S (x,y) = In(1–x² –…

A: Given problem:- Describe the largest set S on which it is correct to say that f is continuous C.…

Q: Let a, b, c, d be real numbers where c and d are not both zero. Let X = {x E R | cx + d + 0}. The…

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Q: By algebraic properties of R, prove that for every x, y € R with y # 0 (-x) y

A: We have to prove that by algebraic property that ∀x,y∈ℝ with y≠0 -xy=-xy

Q: Q3. Let z, y be real numbers such that ry 1 and z #y. Prove that

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Q: Q1- If f is bounded and define on [a, b] then show that m(b – a) < L(P,f)< U(P,f) < M(b – a). Q2-…

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Q: 10. Prove that the set H={[,ne subgroap of GL(2, R). For show that x HxcH but xHx+H. :9 GL(2,R),

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Q: Assume f,g : [a, b] → [c, d] where a < b and c < d. Prove that sup{f(x)+g(x) : x € [a, b]} <…

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Q: Suppose that fo sf,:X-gy and g,-3 YoZ Prove that (Hint : First use ce) to show that gfe g,, and then…

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