Prove that the extreme value theorem doesn’t work over ℚ. That is, find a continuous function, f: ℚ → ℚ defined on a closed and bounded interval I = [a, b] of rational numbers such that f has no maximum on I.

Q: Suppose f'(x0) = 0, f"(xo) = 0, ., f(n-1)(xo) = 0 and f(") (x0) > 0, for a Cn function f. Prove that…

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Q: 2. Prove that the function f : R? → IR defined by f(x, y) = 1- a2 – 4y? is neither injective nor…

A: Given the function f:ℝ2→ℝ defined by fx,y=1-x2-4y2

Q: Suppose that the function f:(a, b) → R is uniformly continuous. Prove that f: (a, b) → Ris bounded.

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Q: Let f : R → R be the function defined as if x 1. Use Theorem 32.5 to prove that f is Darboux…

A: A function is Darboux integrable if for all ε>0, there exists a partition P of [0,2] such that…

Q: Let BCR be the open ball of radius one centered at the origin, i.e. B = {(x₁,...,xn) € R" : x² + +…

A: As per the question we are given a n dimentional open ball B of radius 1 centered at the origin. And…

Q: Suppose that f(x) is a continuous function on [a , b] and F(x)= | f(t) dt a Part 1 of the…

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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: 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.

Q: ded on the y-axi

A: In this question, the concept of maxima and minima is applied. Maxima and Minima The extrema of a…

Q: Let f be analytic in a region G. If f(g) lies on the parabola u=v^2, prove that f is constant in G.

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Q: Which of the following is the hypothesis in the Extreme Value Theorem that sufficiently guarantees…

A: a. f is continuous on [a,b] This is the only hypothesis of Extreme Value Theorem

Q: Suppose that the function f: [0, 1] in R is continuous, f(O) > 0, and f(l) = 0.Prove that there…

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Q: Let f be a real valued function defined on [a, b] . Let f have a local minimum at a point x ∈ (a, b)…

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Q: Without the aid of the Wronskian determine whether the given set of functions is linearly…

A: Given functions: f1x=lnx and f2x=lnx2 To show: Determine whether f1x and f2x are linearly dependent…

Q: Let h: R → R be a continuous function on all of R. If h(x) T, prove that h(T)<0.

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Q: Suppose that f is a function on [a, b] such that f(x) = 0 for all x e [a, b]\T , where T is finite…

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Q: 2. Let f : [a, b] R be a bounded function. Prove that if f is continuous at ro € [a, b), then L(|f])…

A: Here we use the property of continuty.

Q: Prove that if f :V → W is differentiable at a point a E V it must be continuous there.

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Q: xMyn x2+y2 (х, у) 7 (0, 0) f(x, у) %D (х, у) %3D (0, 0)

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Q: Let h: R → R be a continuous function on all of R. If h(x) T, prove that h(Tt) <0.

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Q: Let fbe a continuous function with domain R such that f(1) = 0.05. Prove that fis strictly positive…

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Q: 2. If f is continuous on an open interval containing x, and fchanges its direction of concavity at…

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Q: Let g : R → R be a bounded function and define f : R\{0} →R by 1 f(x) = g(x)+ Using only the…

A: It is given that ;                f : R \ 0 →R  defined by     fx = gx + 1x2 Where,   gx  is a…

Q: prove that any Continuouse function [a,b] en is measurable.

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Q: Let h: R → R be a continuous function on all of R. If h(x) T , prove that h(T)<0.

A: Given that, let h:ℝ→ℝ be a continuous function on all of ℝ. Also given, hx≤0 for all x&gt;π.

Q: Let f be of the class C" in the interval [a.b] and let c be an interior point of this interval with…

A: We have to Show that if f"(c) &lt;0 then f has a local maximum at x=c. We know that A function f has…

Q: If R is a closed region on the xy-plane andf is a continuous function on R, then f has a relative…

A: The given statement is True.

Q: (b) If ƒ is continuous on (a, b) then there is a number c in (a, b) such that f(c) is an absolute…

A: Consider the statement  If f is continuous on (a,b) then there is a number c in  (a,b) such that…

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

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Q: Suppose f'(xo) = 0, f" (xo) = 0, .…, f(n-1)(xo) = 0 and f(") (xo) > 0, for a C function f. Prove…

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Q: If f is continuous and periodic on R, prove that f uniformly contiuous on R.

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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: Use the definition to prove f = (x-1)/(x+1) is uniformly continuous on [0,∞).

A: Given function is fx=x−1x+1. We have to show that fx=x−1x+1 is uniformly continuous on 0,∞. Let us…

Q: Suppose that fn ⇒ f on R and that each fn is a bounded function. Prove that f is bounded.

A: Solution

Q: Suppose f'(x0) = 0, f"(xo) = 0, ..., f(n-1)(x0) = 0 and f(") (xo) > 0, for a C function f. Prove…

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Q: Let h: R R be a continuous function on all of R. If h(x) Tt, prove that h(t)<0.

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Q: Complex analysis

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Q: Suppose that ci < c2 and that f takes on local maxima at ci and c2. Prove that if f is continuous on…

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Q: Prove: If S is compact, and f is continuous on S, then f takes a minimum value some- where in S.

A: Since S is compact and f is continuous, f[S] is a compact subset of R and therefore closed and…

Q: Show (in terms of  −δ) that a function f : [2,7] −→R be defined by f(x) = √x2+1 is uniformly…

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

A: Here we have given that the function fx is diffrentiable at a point  .x=a that is f'x exists.…

Q: Let f be defined on [a, b], if f has a local maximum at a point x E (a, b), and if f'(x) exists,…

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Q: Suppose f is differentiable on (a, b). If c E (a, b) and f' (c) = 0, thenf (c) is either the maximum…

A: Given f is differentiable on (a,b). and c lies in (a,b) such that f'(c)=0.

Q: Let f : (0, 1) → R given by f(x) = 1/x . (i) Prove that f is continuous. (ii) Prove that f is not…

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Q: If x = c is in the domain of f such that f'(c) = 0 and f"(c) > 0, then f has a local maximum at x=…

A: The given problem is solved below in detail.

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: If f has an absolute maximum value over a closed interval [a, b], a, b e R at x = xo, where xo E (a,…

A: Topic:- function

Q: B. Show that the space V = {f : R → R} of functions from R to R is not finite- dimensional.

A: Let V=f:ℝ→ℝ. Let A is set of all polynomial with integer coefficient. We know that every polynomial…

Q: 9. Show that if the analytic function f(z) has a zero of order N at zo, then f(z) satisfying g'(zo)…

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Q: Exercise 11. Prove that if f: E R is continuous at p E E, then there exists & >0 so that f(x) is…

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