Let ƒ : [−1,1] → R be a continuous function. Prove that the sequence (n) defined inductively by Xn 1 2n 1+|xn| Xn+1 = Xn+ ƒ( x₁ = 0, converges. You may use theorems from the lecture as long as you state them explicitly.

Q: Let f(x) be a continuous function. If f(c) is an absolute max or absolute min, then c is unique. O…

A: Given,  fx is a continuous function fc is absolute max or absolute min , then c is unique.  We have…

Q: ,if x>2 Find the correct value of bb that makes

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Q: 3. Consider f(x)=r. For which range of values will the function converge or diverge?

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Q: (x – 2)2 +1 if x 2

A: We have to check continuity of function at x=0 and x=2

Q: 7. Describe the interval(s) on which the function is continuous. Explain why the function is…

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Q: 1. Find the value of c to make the following piecewise function continuous on the the interval (-o,…

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Q: Consider the function f(x) given by the equation. f(x) = 2023 + 2022 + 2023 2021 + 2022 2021 2020 +…

A: We have to find the equation of the line tangent to the graph of f(x) at the point with x-coordinate…

Q: 6. Prove that f(x)=√x is continuous for all x ≥ 0.

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Q: Determine the continuity of the function at the given points. f(x) = [1, x-2, for x=1 at x = 1 and x…

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Q: Let f(t) = 2t² for -4 < t <4 and extend periodically. Compute: f(1) = f(5) = f(-4.5) = f(-8)= f(12)…

A: Periodic function: A function is said to be periodic with period T if f(x+T)=f(x).

Q: Suppose that 0 &lt;ƒ′(x) &lt; 1/2 for all x-values. Show that ƒ(-1) &lt; ƒ(1) &lt; 2 + ƒ(-1)

A: From the given inequality the behaviour of the function can be determined as.

Q: Find the value of the constant c that makes the following function continuous on (-∞, ∞). x² { x = 1…

A: The given function is If function continuous at

Q: Apply the Intermediate Value Theorem to determine whether 8* = x has a solution on the given…

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Q: Discuss the continuity of the function f(x)= [x², x≤1 1+x, x>1 at x = 1.

A: given function To find: continuity of the function at x=1

Q: Suppose f : R→R, then consider a sequence of functions f, : [- 1,1] →R defined by fn(x) := f(nx).…

A: If possible let f is not constant. Then there exist x,y∈-1,1 such that, fx1≠fy1 Let, ε=fx1-fy1 then…