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

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Show that if f is differentiable and f'(x) > 0 on (a, 6), then f is strictly increasing provided…

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Q: Suppose that f [a, b] → R is strictly increasing, i.e., that if x₁ < x₂ then f(x₁) < f(x₂). Explain…

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Q: 1. Use the Cauchy-Riemann conditions to find out whether the following functions are analytic. a.…

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Q: Prove that (r) = tan(r) is uever negative in 0 <r< and show that g(x) = xsinr decreases…

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Q: 5. Let f : R → R be defined by f(x) = { 0 if r € Q, x2 otherwise. Prove that f is continuous at 0…

A: Here we prove the problem using the definition of continuity and a theorem.

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Q: 1. Let the function f: R R be given by 1- cos(x) if x + 0, f(x) := if x = 0. 1 Prove that f is…

A: Differentiablity analysis problem....

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

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Q: · Use the definition of the derivative of complex function to show that f(z) is not dif- ferentiable…

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Q: f(z) = Im z is continuous

A: Let us use the convergence criterion to prove that Im z is continuous. Let us assume that  znn=1∞ be…

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Q: Show by definition that f(x) = |x| : R → R is continuous on R. 2.

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Q: Let f(x) = x²+x+ 2 = Z₂ (x) prove that f(x) irreduciable Poly

A: The given function is  f(x)=x2+x+2.

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

A: we know that for a given function if it is differentiable at given point then it must be continuous…

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

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Q: (a) Let x2 if x is rational f (x) = 0 otherwise. Prove that f is differentiable at 0.

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

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Q: Let f,(x) = x E R. Prove that f 0 uniformly on R. COs nr %3D

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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: Suppose f  is differentiable on an interval I and f'(x) &gt;0for all numbers x in  I except for a…

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

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Q: if f (r) is differentiable prove that f(r)r is irrotational

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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: Use the Cauchy-Riemann Equations to show that f(z) = z is differentiable for 16. all z and find…

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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…

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

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Q: suppose f (x) = Vx (x>0),prove that f is differentiable (0,00) and find f'(x) on

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

Q: Prove that f(x) = tan(x) is never negative in 0 < x < ; and show that g(x) = x=l sin x decreases…

A: We have to prove that f(x)=tan(x) is never negative in the interval 0,π2. Then we have to show that…

Q: prove that if f(x)is derivable at a point say "a" then it is continous at x=a

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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.

Q: Let f = f(x, y), x = s +t and y = s - t. Show that (9- ) - se se as at ax ду

A: It is given that, f=f(x,y), x=s+t and y=s-t. Show that, ∂f∂x2−∂f∂y2=∂f∂s·∂f∂t.…

Q: Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for all r,y R. Prove that for some…

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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: ) 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 that f is analytic on a domain D andVz € D,f'(z) = af(z) for some constant a. Show that f(2)…

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

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

A: Givenf(z)=zz

Q: If f : R + Ris continuous and periodic, prove that f is uniformly continuous.

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Q: Check if the following function is Riemann integrable on [0, 1]: if r is rational f(z) = 1 if z is…

A: Lebesgue's Criterion for integrability- Let f: a,b→ℝ. Then f is Riemann integrable if and only if f…

Q: Suppose that the function f: R → R has the property that -x² < f(x)<x² for all x. Prove that fis…

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Q: Suppose that the function f: R → Ris continuous and that its image f (R) is bounded. Prove that…

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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: