Consider the function f(x) = x^3 + 5x − 4.  a)Use the intermediate Value Theorem to prove that f(x) has a unique root in the interval [0, 1].

Q: (2) (2018A4(b)) Prove from the definition of a limit that the sequence (an) with 2n+1 converges to…

A: Step 1: Step 2: Step 3: Step 4:

Q: 8.) Prove that lim (x₁) = 0 if and only if lin example to show that the the convergence of (Xn).…

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Q: {xn} is a bounded sequence such that all convergent subsequences converge to the same number

A: Every bounded sequence has convergent subsequence.

Q: Suppose that the sequence {n} is such that xn 20 for all n eN and that lim(-1)"xn exists. Show that…

A: We will use Triangle inequality.

Q: In which interval, function f(x)=x3e-x is increasing and concave up? (0, 3(sqrt3)) (3-sqrt3, 3) (3,…

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Q: (6) Let {an} be the sequence defined by an+1 = √6an 2, a₁ = 1. Show that {an} is converger and find…

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Q: let F(x)=f(f(x)) and G(x)=(F(x))^2  You also know that f(5)=10 , f(10)=3,  f prime of 10 = 4, f…

A: The objective of the question is to find the derivative of the functions F(x) and G(x) at x = 5. We…

Q: interval a sxsband a(n) function on a open interval a <x <b. Then there exists ace (a,b) such that…

A: It is problem of mean value theorem

Q: Find number k so that f(z) = 10x + 8 x -1 is continuous on (-0,0). O -2 4 -4

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Q: f(x) = x(x-1) for rational numbers          = 0 for irrational numbers  Prove f is continuous at x…

A: Given function is continuous at x= 0 and x=1. See the attachment.

Q: Q4) Consider the following function f(x) = x³ + 6x² - 4x - 2 a) Find all fixed-point formulations of…

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Q: 6) Determine constants a and b such that f(x) is continuous for all values of x. ax + 3, if x> 5 8,…

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Q: Prove that sequence {an}= {(-1)n} does not have a limit.

A: According to the given information it is required to show that {an}= {(-1)n} has no limit.

Q: (x2+3). Explain why √x4+24 Q7. Consider the function f : [0, 1] → R, x→ (a) f continuous, (b) f is…

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Q: B2. (a) Explain in detail what it means to say that a real sequence (an) is bounded. (b) Prove that…

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Q: Let (an) and (bn) be sequences of real numbers such that (an) converges and (bn) diverges to +∞.…

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Q: Construct a sequence of unbounded functions 9n : [0,1] → R that converges pointwise to a bounded…

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Q: Prove that x = 0 is a discontinuity of each function (a) f(x) = 1 for x

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.