Problem 1. Show that if f is continuous on an interval I, then for any a, b, c E I, | ()dt + f(t)dt. Hint: Assume a < b and consider the four cases: c = a, c = b, c < a < b, a < b < c. (The case a b or a = b.

Q: 7. A function f (x) is continuous on the closed interval [2,5] withf (2) = 17 and f(5) = 17. Which…

A: Rolle's theorem

Q: 12. The function f : [0, 2] → R is continuous on the interval [0, 1] and continuous on the interval…

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Q: Let S = (0, 0), T = (-∞, 2], and W = [-4, 2). For each intersection or union, choose the correct…

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Q: 2. f(x) = is a continuous function but it is discontinuous at x = 3. Explain why X - 3 this is not a…

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Q: 2. The Mean Value Theorem (M.V.T.). First proved by Joseph-Louis Lagrange. 1.) Let f be a continuous…

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Q: . If f : (0, 3) → R is continuous, must its range be an open (possibly infinite) interval?

A: See the detailed solution below.

Q: 1. Suppose that f is a differentiable function and f'(x) ≤ 4 on interval [-5, 2]. If f(2)= 6, then…

A: Suppose that  is a differentiable function and on interval .If , then to determine the smallest…

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

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Q: Assume f is differentiable. Then from the Mean Value Theorem (MVT) it follows that if f(5)0 for some…

A: Plz like ? statement of mean value theorem According to MVT, if a function f is continuous on the…