If f and gare continuous on [a,b]and g(x)>0 for x E[a,b]· 1. Show that there exists CE[a,b] such that | fg=f(c).

Q: Suppose a function a increases on [a, b}, a < xoS b, a is continuous at xo. Let f be a function…

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Q: Suppose f and g are continuous functions on [a, b] and that g(x) > 0 for all x E [a, b]. Prove that…

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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: 3. If f : R R is continuous at ro E R and f(ro) > 0, prove that there exists an interval containing…

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Q: Suppose a function a increases on [a, b}, a < xo sb, a is continuous at xo. Let f be a function…

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Q: Find the values of α and β such that the function f(x) =αcosx + 2 if x < 0  and f(x) = βe3x + αx2…

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Q: If f and gare continuous on [a,b]and g(x)>0 for x E[a,b]- 1. Show that there exists cE[a,b] such…

A: (1) First mean value theorem:- If f:a,b→R and g:a,b→R be both interable on a,b and gx has the same…

Q: Suppose a and b are continuous functions on [a, b] and that g(x) ≥ 0 for all x ∈ [a, b]. Prove that…

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Q: If f(a) = f(b) = 0 and f(x) is continuous on [a, b], then %3D (A) f(x) must be identically zero (B)…

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Q: Let f : R → R, and let x E R. Prove that f is continuous at x if and only if for every e > 0 there…

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Q: Let f : E → R be continuous such that |f(x)|≤|x|, for all x ∈ E. Is f uniformly continuous, justify

A: According to the given information,

Q: and g are continuous on [a,b] and f(x) > g(x) for all x €[a,b] then f(x)dx> ] g(x)dx a

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Q: Let f be differentiable on the interval (a, b) and f'(x) < 4 for all x € (a, b). Then prove that f…

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Q: c =

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Q: If f(z) is continuous at z, then it must be differenttiable at z, ylgn O İhi O

A: If a function f(z) is continuous at z0 , then it may or may not be differentiable at z0. Hence, The…

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: If f and gare continuous on [a,b] and g(x)> 0 for xE[a,b]· 1. Show that there exists cE[a,b] such…

A: We are given that f and g are continuous functions on closed interval [a,b]and g(x)&gt;0 for x∈[a,b]…

Q: 2. Let f(x) = V4 – x for x < 4 and g(x) = x² for all x E R. (a) Give the domain of f · g, ƒ + g, ƒ º…

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Q: Let f be continuous on [a, b] and suppose that f(x) 2 0 for all x e [a,b]. Prove that if L(f)=0,…

A: Given: Let f be continuous on [a,b] and suppose that f(x) is greater than or equal to zero for all x…

Q: If f : R+ → R is a function such that f(x + y) = f(x)f(y) for all x,y ∈ R+. Prove that f is…

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Q: Find all values of c such that f is continuous on (-00, 00). f(r) = { 8 – x², !! x>c

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Q: Suppose f and g are continuous on [a, b] and f f(x) dx = S% 8(x) dr. Prove that there is cE [a, b]…

A: Suppose f and g are continuous functions on [a, b] such that ∫abf(x) dx=∫abg(x) dx We need to prove…

Q: Suppose f and g are continuous functions on [a,b] and that g(x) > 0 for all x E [a, b]. Prove that…

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Q: Find all values of "c" such that the function f(x) is continuous over  ( − ∞ , ∞ ) f(x) = 1 - x^2…

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Q: Let f : R → R, and let x E R. Prove that f is continuous at x i and only if for every e > 0 there…

A: We have to provex that  epsilon delta definition of continuity:

Q: Set f(x) = xª for x ≥ 0 and f(x) = 0 for x < 0. For which values of a is f continuous at 0, and for…

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Q: If a function f(x) is not defined at x-a, then it is not continuous at an open interval (a, b) * O…

A: Given Statement is "False"

Q: Let f be a continuous (and therefore uniformly continuous) function on [a,b). Show that, given e> 0,…

A: Consider the given information.

Q: Suppose f is continuous on [a, b], ƒ'(x) exists at some point x e [a, b], g is defined on an…

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Q: If f(x) and g(x)are continuous on [a,b]and if [f = [8 . Prove that 3ce[a,b] such that f(c)= g(c).

A: Here, we will going to use Intermediate Value  Theorem to prove this result

Q: 4. For a function f:D → R and a point xo in D, define A = {x in D x ≥ xo} and B: {x in D x ≤ xo).…

A: Recall definition of conituity as, Right hand limit = left hand limit = value of function at the…

Q: Suppose that f and g are both continuous functions. Also f, g € [a, b] and that f= fg. Prove that 3c…

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Q: If f is continuous on [a, b] and f(x) 20 for all x E[a, b], then Select one: O True O False

A: Let f(x)=x, x∈2, 3 I=∫23f(x) dx =∫23xdx =x323223 =23x3223 =23332-232 =2333-22

Q: 1 - is continuous at a. f If f(x) 0 for all x in the interval I, and f is continuous at a in I then…

A: 1/f is rational function this is discontinous where either f is discontinous or f is zero  But f is…

Q: Find the value/s of a such that the function f is continuous everywhere.

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Q: Let f be a continuous function on [a, b] such that f(r)a" dx = 0 for every n > 0. Prove that f = 0…

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Q: If f(x) is a continuous function such that it is increasing on [1, 3], then Si f(x)dæ is less than…

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Q: Prove that each of the following functions is uniformly continuous on (0, 1). (a) f(r) = sinz (b)…

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Q: (1- a? f(x) = 1– x² if x E Q, if r ¢ Q. %3D p at which f is continuous. Prove your conclusion.

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Q: Suppose a function a increases on [a, b}, a S xo S b, a is continuous at xo. Let f be a function…

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Q: Suppose a function a increases on [a, b}, a < xo Sb, a is continuous at xo. Let f be a function…

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Q: If f is continuous on [a, b] and f'(x) 0 O f(b) f(b)

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Q: Suppose that f is continuous on [a.b] that f(x) 20 for all xE[a,b] and that f =0 1. Prove that f(x)…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: If f(x) is continuous on a closed interval I, and f(a) and f (b) have opposite signs where a and b…

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Q: If f(x) = is continuous at x=, Find A (Ar? Ax x2

A: Given: fx=x3,x&lt;12Ax2,x≥12 We know that any given functiony=fx is continuous at x=a If,…

Q: 1. Given that f is continuous on a, b|, compare f(b)(b – a) | f(x) dx, and if f is constant on a,…

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Q: If f is a continuous function on R with (1) > 0 and f(4) < 0, then there exists a number c between 1…

A: In this question, we need to check whether the given function is true or false.

Q: If f(x) is continuous with f(a)f(b) < 0, then the e Select one: O an indeterminable number of roots…

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.