Exercise 4: Suppose f is differentiable on (2,6) and continuous on [2,6. Suppose further that 1 < f' (x) < 3 for all E (2, 6). Use the mean value theorem to show that: 4 < f(6)-f(2) < 12.

Q: is uniformly continuous by definition on the interval (-1,1) . 4 - * Show that f(x) =

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Q: Suppose  f is a continuous function defined on a closedinterval [a, b].(a) What theorem guarantees…

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Q: 3. Use the e, d definition to show that f(x) = x² is uniformly continuous on A = [0,3]. %3D

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Q: Is it possible to construct a function f : [0,2] → R that is continuous on [0,2], differentiable on…

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Q: 9 Prove that if m(x) is differentiable on (-∞, 0) and its derivative m'(x) is bounded then m is…

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Q: Verify that the hypotheses of the Mean-Value Theorem (MVT) are satisfied for the function f (x) =…

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Q: Use the definition to prove f =1 is uniformly continuous on [0, 0).

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Q: The function g is continuous on the interval [a, b] and is differentiable on (a, b).Suppose that…

A: g(x) = 0 for 7 distinct values of x in (a, b).Drawn a rough sketch of the case

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A: To prove that given function is continuous. gx,y=xy3x2+y6,   if x,y≠0,00,              if x,y=0,0

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Q: Let f be the function defined by: sin x, x 2 - For what values of x is f NOT continuous?

A: Given function is: fx=sinxx&lt;0x20≤x&lt;12−x1≤x&lt;2x−3x≥2

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Q: 크) Prove that va +b 0. Prove that f is uniformly continuous on [0, 0).

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Q: Is g(x) = ln( x2+2 ) uniformly continuous on the interval (−∞,∞)?

A: given function is                                                        g(x)=lnx2+2 log is defined…

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A: For the given information

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Q: 2. Show that the function f(x) = cos x is uniformly continuous on

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Q: Let f : [0, 1] → [0, 1] be continuous. Show that there exists an element x = [0, 1] such that f(x) =…

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Q: Let f [0, 1] → [0, 1] be continuous. Show that there exists an element x = [0, 1] such that f(x) =…

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Q: If f is continuous on [0, 1], prove that f₁¹ f(x) dx = f₁¹ ƒ(1 − x)dx.

A: This question can be solved using properties of definite integrals.

Q: Prove the following result: A function f that is decreasing on [a,b] is integrable on [a,b].

A: Prove the result that,        "A function f that is decreasing on[a,b] is integrable on[a,b]''

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Q: If f is integrable on [a,b], show by example that F (t) = | f (x) dx is continuous but need not to…

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Q: Let f : [0,∞) → R. Assume that f is uniformly continuous on [0, 1] and on [1,∞). Show that it is…

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Q: Suppose f ≤ 0 , f is continuous on [a, b] and ∫ab f dx= 0. Prove f(x) = 0, ∀ x ∈ [a, b].

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Q: If f : R + Ris continuous and periodic, prove that f is uniformly continuous.

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Q: give an example where f is not derivable at [0, ∞), but is uniformly continuous

A: give an example where f is not derivable at [0, ∞), but is uniformly continuous