: Assume f [a, b] R is continuous and f(x) > 0 for all Є [a, b]. Prove that if So fdx = 0, then f(x) = 0 for all x € [a, b].

Q: sin(3x) az if x > 0 if x = 0 (arctan (¹) if x < 0 f(x)=b

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Q: If f is differentiable at a, then f is continuous at a.

A: If a function fx is differentiable at x=a then function must be continuous at x=a. But a function…

Q: Let f be continuous from [0, 1] to [0, 1]. By posing 6(x) = f(x) – x, show that fhas a fixed point…

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Q: Let f be a real valued function defined on [a, b] that is differentiable on (a, b). Suppose f′(x) ≤…

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Q: 6. Let f be continuous on [a, b]. Show ffdF = f(x) [F(b) – F(a)] for some x € [a, b].

A: This is a consequence of the intermediate value theorem

Q: Show that the function f: C → C given by f(z) = z2 is continuous on C.

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Q: 2. Show that if f : R → R is continuous on R and f(r) = 0 for all r E Q then f(r) = 0 for all x E R.

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Q: Q2: If f(x) and g(x) are continuous on [a,b]and if f = [g Prove that 3c e[a,b] such that f(c) g(c).

A: From the given information. The functions f(x) and g(x) are continuous functions on [a,b]. The…

Q: Let f : [a, b] R be a continuous function on [a, b). Prove that if |f(x)| dx = 0 then f(r) = 0 for…

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Q: Suppose f : [a, b] → R is is continuous and such that fg = 0 for all continuous functions g : [a, b]…

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Q: Let f be continuous on the interval [0, b], where f(x) + f(b − x) ≠ 0 on [0, b].

A: Let I=∫0bfxfx+fb-xdx   I=∫0bfxfx+fb-xdx                     ....... (1) use ∫0afxdx=∫0afa-xdx…

Q: 2: If f(x) and g(x) are continuous on (a,b]and if f =[s . Prove that 3cela,b] such that f(c)=g(c).…

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Q: Assume ƒ is a continuous on [2, 4],, and ƒ(2)= 4. Evaluate ƒ(4).

A: we have been given that f is continuous on [2,4] and f(2)=4. we have to evaluate f(4). the given…

Q: Let f and g be continuous functions on [a, b], and suppose that f(a) g(b). Prove that f(c) = alc)…

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Q: Let f(x) = x. Show directly that f ∊ R[0, 2]

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Q: c. Let f be differentiable on R. Suppose that f'(x) # 0 for every r. Prove that f has at most one…

A: We can explain this statement by 2 explainations given below

Q: 1. "Let fE a[a, b]. For x E (a. b), set F(x)=ff. Prove that F is continuous on [a, b].

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Q: If f is continuous on [a, b] and cE[a, b], then f(x)dx does not exist. Select one: True False

A: Topic:- definite integration of continuous function

Q: Let f, g : [a, b] → [a, b] be two continuous functions that satisfy f ◦ g =g ◦ f. Show that there…

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Q: Suppose f [a, b] → [a, b] is continuous. Prove there exists c € [a, b] such that f(c) = c. (Hint:…

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Q: Let f be an entire function such that Re(f(z)) > 0 for all z E C. Prove that f is a constant.

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Q: Show that the function f:R to R defined by f(x)=5x-12 is continuous using the E-d definition

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Q: Use FTC I to prove that if |f '(x)| ≤ K for x ∈ [a, b], then |f (x) − f (a)| ≤ K|x − a| for x ∈ [a,…

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Q: Prove that the function f : R → R given by f(x) = 3x° – 2x is continuous at 1 using the a.…

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