Prove: If f is continuous on [a,b] and differentiable on nd f'(x) < 0, VE(a, b) then is ƒ is decreasing on [a,b].

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: f(x+y)=f(x)+f(y) and f is continuous at 0. Prove that f is continuous at any a.

A: To show that the continuity at x=0 implies continuity at all x (under the given condition on the…

Q: Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. ||…

A: The given function is f:ℝ2→ℝ defined as: fx,y=y2-x2y4+x2if x,y≠0,01if x,y=0,0 Claim: The given…

Q: 2. Show that f(r) = sin r is uniformly continuous on (0, 00). %3D

A: 2. We need to prove that sinx is uniformly continuous on 0,∞. Definition of uniformly continuous…

Q: Let f : [0, 1] → [0, 1] and f be continuous on [0, 1]. Show that there exists x E [0, 1] such that…

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

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Q: Show that the function f defined by (x, y) = (1, –1) f(x, y) = r? + y (x, y) # (1, –1) x + y is not…

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Q: Prove that f(z)=|z| is nowhere differentiable in C but is continuous everywhere.

A: Given function is fz=z. We have to prove that the function fz is nowhere differentiable in C but is…

Q: Show that the function f defined by 1, (x, y) = (1, –1) | f(x, y) : x2 + Y (x, y) + (1, –1) x + y is…

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Q: If f is continuous at (0, 0) and f(0,0) = 10, then lim f(x, y) = 10. (z,y)-(0,0)

A: A function f(x,y) is said to be continuous at a point (a,b), If it satisfies the following f(a,b)…

Q: Clarify that f f :(0,1) →R defined by f(x): not uniformly continuous. =-

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Q: Suppose f is continuous on [0, 1]. What is lim n→ ∞ 1 S fo 0 f(x) dx?

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Q: Find all values of c such that f is continuous on (-0, 0). 1- x2, f(z) = { – x², x. C = (smaller…

A: Given function is, fx=1-x2   , x≤cx          ,x&gt;c  Since 1-x2 and x are continuous on the…

Q: Prove thait if y=f(x) f(x)=0 when x is a rotional number, then f(x)=0 for all x € R. is continuous…

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Q: 43. Let f(x, y) = Is it possible to define f(0, 0) so x² + y2 * that f will be continuous at (0, 0)?…

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Q: Let f and g be continuous funCHons such that So f(x) dx =9,S f(x) dx = 5, and S g(x)dx= -7 what is…

A: Given, ∫06fxdx=9∫36fxdx=5∫30gxdx=-7

Q: B) If f is differentiable at x, then prove that f is continuous at x Is the converse true? Why

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Q: B) Let f:[-2,2]R. then show that the function xsin -) for x 0, f(x) = for x = 0, is uniformly…

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

A: Consider the given function:

Q: Let f be a function defined on R satisfying If(z) - f(y)| < 히 - J), V,yER. a. Show that f is a…

A: Given that f be a function defined on ℝ satisfying…

Q: Let f be continuous on [a,b] and differentiable on (a,b). Prove that if f'(x)>0 for all x=(a,b) then…

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Q: Let f and g be continuous on the closed interval [0, 5]. Suppose that J f(x)dx = A, f(x)dx = B, and…

A: Given: ∫05f(x)dx=A∫03f(x)dx=B∫53g(x)dx=C…

Q: Prove that if f is continuous on [a, b] and F(x) = ſx ƒ (t)dt α then F is continuously…

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Q: If f is continuous on [0, 1] such that x* f(x) dx=0, for n0, 1, 2, 3,.. then show that f(x)=0 on [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

Q: Show that the function f defined by 1, (x, y) = (1, –1) %3D f(x, y) = { a2 + y (x, y) # (1, –1) x +…

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Q: Suppose that f and g are continuous on I = [a, b], and they are differentiable on (a, b) with f'(x)…

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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

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

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Q: Define f(0,0) in a way that extends f to be continuous at the origin. 6x² -x²y² + 6y² x² +y? f(x,y)…

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Q: Σ sin k+1

A: to show that the given series ∑k=1∞1ksinxk+1 converges uniformly on R to a differentiable function f…

Q: 2. Let f be a differentiable continuous function on [a, b] and suppose f' is continuous on [a, b].…

A: In this question we will use basic of calculus i.e properties of integration. We will use…

Q: Show that the function f(x) = csc(x) is not uniformly continuous on (0, pi/2]

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Q: Suppose that f has continuous derivative and is twice differentiable in R. Suppose that f(a) = f(b)…

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Q: b) Show that the function sin for æ e (0, 1] { f (x) = for æ = 0 is absolutely continuous on [0, 1].

A: Function is called continuous function if the left and right hand limit values at each and every…

Q: 1. Suppose f(r) is a continuous function on r E [a, b]. Prove f(x)dx < Slf(x)| dx.

A: According to our guidelines we can solve only first question and rest can be reposted.Thank you

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]''

Q: 5.Prove that: if f is continuous at x = 0, then xf (x) is differentiable at x = 0.

A: We want to prove the given result.

Q: Consider a function ƒ : [0, 1] → R defined by f(x) = x(x³ - 1). (a) Can we apply Rolle's Theorem to…

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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…

A: We need to show by an example that if f is integrable on [a, b], then F(t) (as defined in the…

Q: 2. Let f (x) = cos-. Show that f is NOT uniformly continuous on (0,1).

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Q: What is the continuous extension of f(x) = at point x-1. x -1 x2-1' %3D x +x-2 find dy dx

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Q: Let f(x) be continuous on [0, oo) and twice differentiable on (0, co). If f'(x) > 0 and f"(x) 1,…

A: Let f(x) be a continuous function on [0,∞) and differentiable at 0,∞. The given double derivative…

Q: 1. Estimate the limit of the signum function s(x) as x approaches zero from the right. (-1, x0

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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: Prove f (x) 1+ mx is continuous at x = 1, where m e R

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Q: 3. Let f(r) = r sin() if z #0, and f(0) = 0. Show that f is continuous at r= 0 by using e- d…

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Q: Consider the sequence of functions {fn} defined on the interval (1, 0), where fn (x) 1 Vx E (1, 0),…

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