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

Q: If we apply Rolle's Theorem to the function f(x) = 2x – 12x – 3 on the interval [1, 5], how many…

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

Q: If f is continuous but not differentiable at x = c then there is a corner on its graph at (c, f(c)).…

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Q: For what value of the constant c is the function f continuous on (−∞, ∞)? f(x) =    cx2 + 6x…

A: fx=cx2+6x     if x<3x3-cx       if x≥3 fx is continuous for all x<3 & x>3, because it…

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

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Q: Show that if f: A-> B is continuous and A is compavt then f(A) is also compact

A: Let , f : A →B be continuous function. Here , A is compact set. We need to prove that , f(A) is also…

Q: If f(x) is continuous on the interval [a, b] and a <c < b, | f(x) dæ is equal to then

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Q: Show that f(r) = O(|x – xo|*) and g(x) = o(]x – roP) imply f. g(x) = o(|x – xo]k+i). - -

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Q: Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c…

A: The Intermediate Value Theorem that states if a function f(x) is continuous and k is     any…

Q: If we apply Rolle's Theorem to the function f(x) = 2x² – 16x – 5 on the interval [2, 6], how many…

A: Given,          fx=2x2-16x-5 on the interval 2,6.We know that,         Rolle's theorem states that…

Q: If for a real continuous function f(x), f(a)*f(b)<0, then in the range of [a,b] for f(x)=0, there is…

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Q: If f(x) is continuous in the interval [-2, 1], ƒ(−2) = −4, and f(1) = -1, can we use the…

A:

Q: If  and f is continuous, then  f must have at least one zero between  a and b. (Assume a ≠ b.)

A: Case1: Whenfx=0 , then ∫abfxdx=0 is obvious.

Q: f(x)={x^2−9x≤c, 2x−10 x>c } i

A: Consider function f(x)=x2-9           x≤c2x-10        x&gt;c Now  limx→c+ f(x)=limx→c+…

Q: f(ix, λγ) - λκ. f(x, y)

A: Given: fλx,λy=λkfx,y

Q: D. Prove that f(x) = |x - 6| is continuous at all points c in R.

A: Given function is fx=x−6. We have to prove that the function fx=x−6 is continuous at all point c in…

Q: If f: X →Y is continuous and E C Y is closed, then f-(E) must be: - Closed Open None of these

A: The given function: f:X→Y is continuous. That is, f(X)=Y is continuous. Then, E is the subset of Y…

Q: Let f and g represent differentiable functions such that f ″ ≠ 0 and g″ ≠ 0.

A: Concavity of a function can be determined using the second derivative of the function. If the second…

Q: If f(g(x)) is continuous at 0, then g(x) is continuous at 0  true or false

A: Given that If f(g(x)) is continuous at 0, then g(x) is continuous at 0

Q: If f is continuous on [a, b], then the horizontal line y = k for every k between f (a) and f (b)…

A: If function 'f'  is continuous on closed interval [a, b], and if f(a)≠ f(b), then for any number k…

Q: Suppose that  ff"' is continuous and f'(c)  =f"(c) =0 , but f"'(c) &gt;0. Does f have a local…

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Q: If a function f(x) is not defined at x-c, then it is not continuous at a close interval [a, c). * O…

A: We know that function is continuous at c if the 1) f(c) is defined 2)limit of f(x) exist, Here…

Q: For what values of a and b is -2, xS-1 f(x) = ax - b, -1 <x<1 3, continuous at every x?

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Q: Prove that f (x) = x-12 is : (i) not sectionally continuous in every finite interval in the

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Q: Can Rolle's Theorem be applied to f(x) = 1/(1-x) on interval [0,3]? If yes, what is c in (0,3) such…

A: here we can't apply the rolles Theorem

Q: Suppose that f : R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the 8–neighborhood…

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Q: Consider the function f with domain (-2, ): 3|x2 – 1| 2x2 + 2x – 4 if – 2< x < 1 f (x) = 3 if x = 1…

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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: Suppose f and g are differentiable functions on R such that,

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Q: If f : [0, 3] → R is continuous not constant, then its range be a closed, bounded interval?

A: Yes true.

Q: Suppose f'(x) = c for all x, where c is a constant. Show that, for some constant d, f(x) = cx + d.

A: suppose f'(x) =c for all x, where c is constant .Show that for some constant d, f(x)=cx+d

Q: If a function f(x) is not defined at x-a, then it is not continuous at x-a.

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Q: Suppose f(z) is analytic for |z| &lt;= 1, and |f(z)| &gt;=1. If f(0) =1, prove that f(z) is…

A: To prove that the given conditions on the analytic function force it to be the constant function…

Q: If f : [0, 3] → R is continuous, must its range be a closed interval?

A: Given data: The expression for the function is f : [0,3]→R. The continuous image of the open bounded…

Q: show that f,given by f(x) x +0 )is continuous on R-{0}

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Q: Given that f(z) = z where does f'(z) exist

A: The given function is,  f(z) = z. We have to find those points where f'(z) exists.

Q: If f(x) is differentiable at c, then there exists d > 0 such that f(x) is continuous in (c – 8, c+…

A: We have to define it is true or not

Q: On which of the following interval is f(x) S O (0, +∞0) ○ (2, +∞) O [2, +∞0) 0 (-∞, +∞0) 1 x-2…

A: We will the fact that root value should be positive.

Q: Explain why the following function satisfies the hypotheses of Rolle's Theorem on the given…

A: As the function is a polynomial function so it is differentiable on (-2,-1) and continuous on…

Q: Suppose that f(x) is continuous and differentiable, f(2) = -2, f(6) = -6, and f(x)dx =-10. Compute ,…

A: Topic:- Functions

Q: If f is a function satisfying |f (x) – f(y)| < (x – y)? - for all x and y, prove that f is constant.

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Q: Determine if f(x)=x² is uniformly continous on the interval I = (-∞, +)

A: See the attachment.

Q: f(x)=(x-6)³+4, [4,7],f(c)=3 3

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Q: If f(x) = then f (x) at z = 1 is

A: Given that :-         f(x)=1x         f'(x)=? at x=1

Q: If the function f (x) = } 6x – 4, x5 is continuous at x =5, then the value of k is %3D %3D O a. 10 O…

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Q: Prove: f(x)=5x y is continuous at x=2

A: A function fx is continuous at a point x=a, if limx→afx=fa Given function is

Q: If f(2) = 14, f' is continuous, and f'(t)dt = 23, what is the value of f(5)=? %3D

A: Given :

Q: Suppose f(x) is a bounded real - valued function on [a.b) such that fela,b] . Does it follows that…

A: A function is bounded in [a,b] if it can't achieve value beyond a and b. i.e.maximum acheivable…

Q: Determine the interval(s) where f(x)-In (5x+10) is continuous. It is continuous in the interval

A:

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.