Let f be differentiable on R. Suppose that f'(r) # 0 for every r. Prove that f has at most one real root.

Q: Show that f (x) = 1 is not uni form continuous on (0,3) I-3

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

A: Given data is: Let f be differentiable on R. Suppose that f'x≠0 for every x. To prove f has at most…

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Q: Prove that f(x) =√x is uniformly continuous on [0,∞).

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Q: If ƒ is differentiable at a, must ƒ be continuous at a?

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Q: Prove that if f is continuous, then f(t)(x-t) dt = f(u) du ) dt

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Q: Let f : R → R with f(x) = 3x² + 2. Prove that f is differentiable at the point –1 %3D

A: f(x) =3x2+2 f'(x) =limh→0f(-1+h) -f(-1)h        =limh→03(-1+h)2+2-3-2h        =limh→03+3h2-6h -3h…

Q: Prove that f (z) =zz is nowhere analytic

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A: See the attachment

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Q: 2. Determine if a function f : [0, 2m] →R defined by f(r) = sin 2x is continuous. %3!

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

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Q: if x = 0 f (x) = %3D x* sin(-) otherwise

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A: Function f(x) = √x (x&gt;0)  We know f(x) =Xn

Q: Consider the function f: (-2,0) → [0, 0) defined by f (x) = Vx + 2. Prove that f is suriective.

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Q: Prove that the function z is continuo:as everywhere but nowhere differentiable except at origin.

A: The given problem is solved below in detail.

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Q: ) For a, 3 ER and B > 0, let f(x) ={} x° sin(|x|-#), when x # 0 when x = 0 0, ange of a and B (and…

A: Continuity and differentiability

Q: If f is analytic and has a pole at z = a, then f(2)|| → 0 as z → a in any manner.

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Q: Suppose f ≤ 0 , f is continuous on [a, b] and ∫ab f dx= 0. Prove f(x) = 0, ∀ x ∈ [a, b].

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Q: Show that the function f(z) = z = is differentiable but not analytic at z = 0

A: Givenf(z)=zz

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Q: Suppose that F(x, y, z) = 0 defines z as a function of x and y and F is differentiable. Prove that…

A: This is a problem of the chain rule and Implicit function theorem.

Q: Let f : (0, ∞) → R be defined by f(x) = ln x. Prove that f is 1-1.

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: