1. Show that the function, defined for x E R by f(x) = 1 n2 1 is uniformly continuous on R.

Q: Exercise 4. Let f e BV ([0, a)) for some a > 0. Define :| S(1)dt, r € (0, a], F(x) = a = 0. Show…

A: Since it is given that f∈BV([0,a]), then according to Jordan decomposition theorem mentioned there…

Q: 7. For continuous functions on [0,1], 1/2 | f(x) | dx < | f(x) |² dx Hint: Recall the Cauchy-Schwarz…

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Q: If f isa differentiable function satisfying f(0) = 0 and f (0) = 2, then 1- cos( f (x)) lim x-0 A) 0…

A: The given limit is, limx→01-cos(f(x))x2. Note that, the given limit is of the form 00. Apply L…

Q: II. Find all values of a and b that make -a -x, x<-2 f(x) = {bx² + 1, Vx + b, -2 Sx<1 x21 continuous…

A: Given:            fx=-a-x,       x&lt;-2bx2+1,     -2≤x&lt;1x+b,       x≥1 Continous everywhere:  at…

Q: AP1) Prove that f: E → R; x x² is uniformly continuous if E is bounded. AP2) Prove that f : R R; x x…

A: Definition: A function fx said to be uniformly continuous, if for every ε&gt;0, there exists δ&gt;0…

Q: Is the proof correct? Either state that it is, or circle the first error and explain what is…

A: The given proof is correct.

Q: Q2/Verify Lagrange's mean value theorem for the function F(x) = Vx2 – 4 in the interval [2,4]…

A: Given (2) The given function is Fx=x2-4 and the interval is 2,4. (3) The given expression is…

Q: 4. Choose and work either a) or b), NOT BOTH. (a) Show that the function f (x) = sin z for x €R is…

A: In the question given that we have to solve any one of them. Let us solve the first one.

Q: Is it possible to construct a function f : [0,2] → R that is continuous on [0,2], differentiable on…

A: According to question given :f: 0,2→Rcontinuous on 0,2Differentiable on 0,2f(0)=0, f(2)=-3And…

Q: consider the uniformly continuous function f(X)=√x [0,1]. for a given ε>0 , what is the largest…

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Q: f(x)dx converges then lim,f(x)=0

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Q: 2. Is the proof correct? Either state that it is, or circle the first error and explain what is…

A: The following small changes are made in the proof. Let ε&gt;0 be given. Since f is uniformly…

Q: x² + 1, x 1 Let f(x) = Is function f continuous at x = 1? Is functionf differentiable at x = 1?

A: Given :-  f(x) = x2+1 when x&lt;1  f(x) = 4-2x when x≥ 1  To find:- Check function continuous or…

Q: Suppose that f : [a,b] → R is continuous and with continuous derivative f'. Suppose also that f(a) =…

A: Suppose f:[a,b]→R be a continuous and continuously differentiable such that f(a)=f(b)=0 and…

Q: (c) A non-constant function f : [0, 1] → R such that f(x) > 0 for any r e [0, 1] but | f(x)dx = 0.…

A: Let us consider a function f:0,1→ℝ such that fx≥0 for all  x∈0,1.  Suppose fxn be a sequence of…

Q: 3. f(x) = Vx - 5 is continuous on the interval [5, 0) %3D

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Q: (c) A non-constant function f : [0, 1] →R such that f(x) > 0 for any r € [0, 1] but | f(x)dx = 0.…

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Q: Draw the graph of a continuous function f(x) where f(-2) = 0, f(2) s no value c E (-2, 2) such that…

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Q: 1) All continuous f:[0, e function oro impanding a2 x³f (x*) dx = Sxf(xldx a) show +that 6)*>o…

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Q: A continuous function f satisfies the equation 1 f (x) = x – t+ 2 f(t) dt 0, For all x e [-1, 3].…

A: We have to use leibiniz rule of  differentiating definite integral  d∫f(x)g(x)h(t)…

Q: (c) A non-constant function f : [0, 1] → R such that f(x) > 0 for any x e [0, 1] but | f(x)dx = 0.…

A: In the given question we have to find the  function the function f(x) s.t f:[0,1]→ℝ s.t…

Q: 55. Let F be a function given by Sx2 + 1, for x s 2, 2x + 1, for x >2. F(x) = a) Verify that F is…

A: We have Fx=x2+1 ,  for x≤22x+1 , for x&gt;2 We have to verify   (a) Function is continuous at x=2…

Q: (c) A non-constant function f : [0, 1] → R such that f(x) > 0 for any x e [0, 1]I f(x)dx = 0. Give a…

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Q: V4 - a2 Given f(x) and g(x) = |x|, fogis continuous on the interval %3D 이-2,-1] U [1,2] o[-2, 1) U…

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Q: 1. Use the properties of continuous functions to explain why f(x) =x cos(x) +1 is a continuous…

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Q: function f(x) such that f'(x) = (2x – 1)e²“. O Find the interval/s on which ƒ is increasing /…

A: Explanation: Given that,                          find the critical points to put the f'(x) = 0…

Q: 2 State why Rolle’s Theorem does not apply to the function f (x) on the interval [-2,0] (x+1)² f is…

A: Given query is to find the correct option why Rolle's  theorem can not be applied on f(x).

Q: Let X 2 0 be continuous. Show that E[X] = So° (1 – F(x)) dx.E[X] = S,° (1 – F(x)) dx. %3D

A: Let X be a continuous variable lies in interval [-∞,∞] with its probability density function f(x).…

Q: 2. Show that the function, defined by f(x) : is NOT uniformly continuous on its x2 domain D(f) = {r…

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Q: Suppose a continuous function f is concave up on (- 00,0) and (0, 00). Assume f has a local maximum…

A: Since you have asked multiple questions, we can answer only one question. Please post again to get…

Q: 25. Find the constant(s) c for which the function f(x) is continuous on (-∞,+ ∞) (c? — х2 , х<2 12…

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Q: 2. Is the proof correct? Either state that it is, or circle the first error and explain what is…

A: The proof is, Let ε&gt;0 Since f is  uniformly continuous on (0,1] there exists δ1&gt;0 and…

Q: (3) Let f : R –→ R be a differentiable function everywhere. Prove that if f (x) = 0 has n solutions,…

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Q: Q2/ let f (x) x2-4x +7, then f is continuous on [0,4] and differentiable on (0,4) find the value ce…

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Q: (-x^2, x < a Vx, x z a Find the value of a such that f (x) = f(x) = is continuous everywhere.

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Q: a. Let f(x) be a function satisfying |f(x)| < x² for-1 sxs 1. Show that f is differentiable at x = 0…

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Q: Exercise (b) Use Taylor's Inequality to estimate the accuracy of the approximation f= Tp(x) when x…

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Q: true or false with reason a)if f is continuous in[a,b] then integrationba x *f(x)dx= x…

A: a) False Given that fx is continuous in [a,b] .Then ∫abx fxdx=x∫abf(x)dx............(1). Note that…

Q: Newton-Raphson method will always converge to a solution for f(x)= 0 on the interval a s x s b if…

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Q: True or false: Let f(x) be a continuous function. Suppose that f(x)dx converges for some real number…

A: We have used some properties of integration.

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: Q2/ let f(x) = x2-4x + 7, then f is continuous on [0,4] and differentiable on (0,4) find the value…

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Q: 3. Let f [2,3] → {1,3} be a continuous function such that the equation f(x) - x = 0 has a solution a…

A: 3 Give that f:2,3→1,3 be a continuous function such that the equation fx-x=0 has a solution x in…

Q: 1-x², x 2 at x = 2.

A: Given, fx=1-12x2, x&lt;23,    x=2x-2-1,   x&gt;2

Q: Let f (x) = ex – e2x. Find all extreme values (if any) of f on the interval [0, 1]. Determine at…

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Q: True (T) or False (F). If c is a critical number of f(x), then f '(c) = 0. If f(c) exists and f '(c)…

A: 1. If c is critical number of fx then f'c=0TRUE  (Critical point definition)2.If fc exist and f'c=0,…

Q: If f is positive-valued continuous decreasing function and lim - S(x) = 0 then J F(x)dx diverges…

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Q: true or false with reason a)if f is continuous in[a,b] then integrationba x f(x)dx= x…

A: a). False. The result of doing this would give you a function of x, but the definite integral is a…

Q: 2. Is it possible for a function f(x, y) to satisfy both f(1, 2) = 5 and f(0,0) = 5? If yes, gi- an…

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Q: The Fundamental Theorem of Calculus (Part 1) states that if f(x) is a continuous function on an…

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